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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 Coordination 1.5 Focal Points Notes Problems 1 xxx February 2, 2000. December 12, 2003. 24 March 2005. Eric Rasmusen, firstname.lastname@example.com. http://www.rasmusen.org/GI Footnotes starting with xxx are the author's notes to himself. Comments are welcomed. vi 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 Strategic 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 Nuisance 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 Switching 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 Information 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 Kemper August 6, 2003. 24 March 2005. Eric Rasmusen, email@example.com; 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 Introduction, 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 xv 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 reorganized 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 website), 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 Hidden 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 Example 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 Equilibrium: 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 answers, 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: mathematical 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 mathematical 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 firstyear 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 improving 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 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 competitor'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 understand 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 Organization, 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, Oxford: 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. xix 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 overshadowed 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 University 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. xx 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: John Wiley 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 Theory, 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 edition, 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 University Press. 284 pages. With unusual applications (chapters on macroeconomics, trade policy, and environmental economics) and lots of exercises with answers. xxi 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, Oxford: 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 Mathematical 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 problems and solutions, with much explanation and special attention to evolutionary biology. 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 economics. 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 reasons should email me at firstname.lastname@example.com; send me snailmail at Eric Rasmusen, Department of Business Economics and Public Policy, Kelley School of Business, Indiana 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 formatted 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 email@example.com. 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 accounting), Paul Cheng, Luis Fernandez (Oberlin economics), David Hirshleifer (Ohio State finance), Jack Hirshleifer (UCLA economics), Steven Lippman (UCLA management science), 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 (Appalachian 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-Shin Wei 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 (University 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 Neumann & 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 terminology 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 1 July 24, 1999. May 27, 2002. Ariel Kemper. August 6, 2003. 24 March 2005. Eric Rasmusen, email@example.com. 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 methodology of economics. In the past, macroeconomists started with broad behavioral relationships 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 endowments 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 utility subject to the constraints imposed by their information, endowments, and production functions. This is exactly the paradigm of game theory: the modeller assigns payoff functions 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 maximization 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. Exemplifying 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 assumptions 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 mathematics is required to pin down the modeller's thoughts. Exemplifying theory treads a middle path between mathematical generality and nonmathematical vagueness. Both alternatives 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 technique, 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) permitted 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 enables 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 consistency 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, firstname.lastname@example.com. 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 individual 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 1 Once 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 pseudoplayers 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}. We will 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 Enter OldCleaner Low price High price -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 Enter OldCleaner Low price High price -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 probability 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 actions 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 NewCleaner 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: 2 I 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 (2) πi (sdi , s−i ) < πi (s0i , s−i ) ∀s−i . 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 s00i . 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 dominant 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 separately. 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: Conf ess 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 3 Another 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 crimefighting 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 Conf ess 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 noncooperative 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 monopsonist. 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 4 If 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 paretodominate 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 Conf ess, 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 exactly 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 s00i 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 s00i such that πi (s00i , s−i ) ≥ πi (s0i , s−i ) ∀s−i , and πi (s00i , s−i ) > πi (s0i , s−i ) for some s−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 Kenney decides that Imamura will pick North because it is weakly dominant, so Kenney eliminates "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 important 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 simultaneously; 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 equilibrium 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 dominance 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 deletion 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 c2 c3 c1 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 perfectness". 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 P ress 26 the panel and W ait by the dispenser at the other end. Table 5: Boxed Pigs Small Pig Press Wait 5, 1 → 4, 4 ↓ ↑ 9 , −1 → 0, 0 Press Big Pig Wait 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, (P ress, W ait), 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 (P ress, W ait) 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 P ress, 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 W ait, 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 (P ress, W ait) 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 5 This 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 (P ress, W ait). 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 equilibrium strategy and some other strategy. Every dominant strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium 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). Deny Deny Table 6: The Modeller's Dilemma Row 0, 0 l Column Conf ess ↔ Conf ess 0 ,-10 → Payoffs to: (Row, Column) −10, 0 ↓ -8 , -8 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 o