Game Theory for Applied Economists Robert Gibbons Princeton University Press Princeton, New Jersey Contents 1 Static Games of Complete Information 1 1.1 Basic Theory: Normal-Form Games and Nash Equilibrium. . . . . . . . . . . . . . . . . . . . . 2 1.1.A Normal-Form Representation of Games. 2 1.1.B Iterated Elimination of Strictly Dominated Strategies . . . . . . . . . . . . . . . . . . . 4 1.1.C Motivation and Definition of Nash Equilibrium 8 1.2 Applications . . . . . . . . 14 1.2.A Coumot Model of Duopoly 14 1.2.B Bertrand Model of Duopoly 21 1.2.C Final-Offer Arbitration . . . 22 1.2.D The Problem of the Commons 27 1.3 Advanced Theory: Mixed Strategies and Existence of Equilibrium . . . . . . . . 29 1.3.A Mixed Strategies . . . . . . . . . 29 1.3.B Existence of Nash Equilibrium . 33 1.4 Further Reading 48 1.5 Problems 48 1.6 References ... 51 2 Dynamic Games of Complete Information 55 2.1 Dynamic Games of Complete and Perfect Information. . . . . . . . . . . . . . . . 57 2.1.A Theory: Backwards Induction . . . 57 2.1.B Stackelberg Model of Duopoly. . . 61 2.1.C Wages and Employment in a Unionized Firm 64 2.1.D Sequential Bargaining . . . . . . . . . . 68 2.2 Two-Stage Games of Complete but Imperfect Information. . . . . . . . . . . . . . . . . . . . 71 viii CONTENTS 2.2.A Theory: Subgame Perfection . 71 2.2.B Bank Runs . . . . . . . . . . . 73 2.2.C Tariffs and Imperfect International Competition 75 2.2.D Tournaments . 79 2.3 Repeated Games . . . 82 2.3.A Theory: Two-Stage Repeated Games 82 2.3.B Theory: Infinitely Repeated Games 88 2.3.C Collusion between Coumot Duopolists . . 102 2.3.0 Efficiency Wages . . .. ..... . 107 2.3.E Time-Consistent Monetary Policy 112 2.4 Dynamic Games of Complete but Imperfect Information. . . . . . . . . . . 115 2.4.A Extensive-Form Representation of Games 115 2.4.B Subgame-Perfect Nash Equilibrium . 122 2.5 Further Reading . 129 2.6 Problems . . . . . . . 130 2.7 References . 138 3 Static Games of Incomplete Information 143 3.1 Theory: Static Bayesian Games and Bayesian Nash Equilibrium . ......... .......... . 144 3.1.A An Example: Cournot Competition under Asymmetric Information . . . . . . . . .144 3.1.B Normal-Form Representation of Static Bayesian Games ............ . .146 3.1.C Definition of Bayesian Nash Equilibrium .149 3.2 Applications . . . . . .152 3.2.A Mixed Strategies Revisited 152 3.2.B An Auction . . . . . .155 3.2.C A Double Auction . .158 3.3 The Revelation Principle 164 3.4 Further Reading .168 3.5 Problems .169 3.6 References .172 II Dynamic Games of Incomplete Information 173 4.1 Introduction to Perfect Bayesian Equilibrium. . 175 4.2 Signaling Games . . . . . . . . . . . . . . . 183 4.2.A Perfect Bayesian Equilibrium in Signaling Games. . . . . . . . . . . . . . . . . . . .. .183 Contents ix 4.2.B Job-Market Signaling . . . . . . . . . . . . . .190 4.2.C Corporate Investment and Capital Structure . 205 4.2.0 Monetary Policy . . . . . . . . . .208 4.3 Other Applications of Perfect Bayesian Equilibrium ............ . .210 4.3.A Cheap-Talk Games. . . . . . . . .210 4.3.B Sequential Bargaining under Asymmetric Information. . . . . . . . . . . . .218 4.3.C Reputation in the Finitely Repeated Prisoners' Dilemma . . . . . . . . . .224 4.4 Refinements of Perfect Bayesian Equilibrium . .233 4.5 Further Reading .244 4.6 Problems . .245 4.7 References .253 Index 257 Preface Game theory is the study of multiperson decision problems. Such problems arise frequently in economics. As is widely appreciated, for example, oligopolies present multiperson problems - each firm must consider what the others will do. But many other ap plications of game theory arise in fields of economics other than industrial organization. At the micro level, models of trading processes (such as bargaining and auction models) involve game theory. At an intermediate level of aggregation, labor and finan cial economics include game-theoretic models of the behavior of a firm in its input markets (rather than its output market, as in an oligopoly). There also are multiperson problems within a firm: many workers may vie for one promotion; several divisions may compete for the corporation's investment capital. Finally, at a high level of aggregation, international economics includes models in which countries compete (or collude) in choosing tariffs and other trade policies, and macroeconomics includes models in which the monetary authority and wage or price setters interact strategically to determine the effects of monetary policy. This book is designed to introduce game theory to those who will later construct (or at least consume) game-theoretic models in applied fields within economics. The exposition emphasizes the economic applications of the theory at least as much as the pure theory itself, for three reasons. First, the applications help teach the theory; formal arguments about abstract games also ap pear but playa lesser role. Second, the applications illustrate the process of model building - the process of translating an infor mal description of a multiperson decision situation into a formal, game-theoretic problem to be analyzed. Third, the variety of ap plications shows that similar issues arise in different areas of eco nomics, and that the same game-theoretic tools can be applied in xii PREFACE each setting. In order to emphasize the broad potential scope 01 : the theory, conventional applications from industrial organization. largely have been replaced by applications from labor, macro, and other applied fields in economics.l We will discuss four classes of games: static games of com plete information, dynamic games of complete information, static games of incomplete information, and dynamic games of incom plete information. (A game has incomplete information if one player does not know another player's payoff, such as in an auc tion when one bidder does not know how much another bidder is willing to pay for the good being sold.) Corresponding to these four classes of games will be four notions of equilibrium in games: Nash equilibrium, subgame-perfect Nash equilibrium, Bayesian Nash equilibrium, and perfect Bayesian equilibrium. Two (related) ways to organize one's thinking about these equi librium concepts are as follows. First, one could construct se quences of equilibrium concepts of increasing strength, where stronger (Le., more restrictive) concepts are attempts to eliminate implausible equilibria allowed by weaker notions of equilibrium. We will see, for example, that subgame-perfect Nash equilibrium is stronger than Nash equilibrium and that perfect Bayesian equi librium in turn is stronger than subg ame-perfect Nash equilib rium. Second, one could say that the equilibrium concept of in terest is always perfect Bayesian equilibrium (or perhaps an even stronger equilibrium concept), but that it is equivalent to Nash equilibrium in static games of complete information, equivalent to subgame-perfection in dynamic games of complete (and per fect) information, and equivalent to Bayesian Nash equilibrium in static games of incomplete information. The book can be used in two ways. For first-year graduate stu dents in economics, many of the applications will already be famil iar, so the game theory can be covered in a half-semester course, leaving many of the applications to be studied outside of class. For undergraduates, a full-semester course can present the theory a bit more slowly, as well as cover virtually all the applications in class. The main mathematical prerequisite is single-variable cal culus; the rudiments of probability and analysis are introduced as needed. 1 A good source for applications of game theory in industrial organization is Tirole's The Theory of Industrial Organization (MIT Press, 1988). Preface xiii I learned game theory from David Kreps, John Roberts, and Bob Wilson in graduate school, and from Adam Brandenburger, Drew Fudenberg, and Jean Tirole afterward. lowe the theoreti cal perspective in this book to them. The focus on applications and other aspects of the pedagogical style, however, are largely due to the students in the MIT Economics Department from 1985 to 1990, who inspired and rewarded the courses that led to this book. I am very grateful for the insights and encouragement all these friends have provided, as well as for the many helpful com ments on the manuscript I received from Joe Farrell, Milt Harris, George Mailath, Matthew Rabin, Andy Weiss, and several anony mous reviewers. Finally, I am glad to acknowledge the advice and encouragement of Jack Repcheck of Princeton University Press and finandal support from an Olin Fellowship in Economics at the Na tional Bureau of Economic Research. Game Theory for Applied Economists Chapter 1 Static Games of Complete Information In this chapter we consider games of the following simple form: first the players simultaneously choose actions; then the players receive payoffs that depend on the combination of actions just cho sen. Within the class of such static (or simultaneous-move) games, we restrict attention to games of complete information. That is, each player's payoff function (the function that determines the player's payoff from the combination of actions chosen by the players) is common knowledge among all the players. We consider dynamic (or sequential-move) games in Chapters 2 and 4, and games of incomplete information (games in which some player is uncertain about another player's payoff function-as in an auction where each bidder's willingness to pay for the good being sold is un known to the other bidders) in Chapters 3 and 4. In Section 1.1 we take a first pass at the two basic issues in game theory: how to describe a game and how to solve the re sulting game-theoretic problem. We develop the tools we will use in analyzing static games of complete information, and also the foundations of the theory we will use to analyze richer games in later chapters. We define the normal-form representation of a game and the notion of a strictly dominated strategy. We show that some games can be solved by applying the idea that rational players do not play strictly dominated strategies, but also that in other games this approach produces a very imprecise prediction about the play of the game (sometimes as imprecise as "anything could 2 STATIC GAMES OF COMPLETE INFORMATION happen"). We then motivate and define Nash equilibrium-a so lution concept that produces much tighter predictions in a very broad class of games. , In Section 1.2 we analyze four applications, using the tools developed in the previous section: Cournot's (1838) model of im perfect competition, Bertrand's (1883) model of imperfect com petition, Farber's (1980) model of final-offer arbitration, and the problem of the commons (discussed by Hume [1739] and others). In each application we first translate an informal statement of the problem into a normal-form representation of the game and then solve for the game's Nash equilibrium. (Each of these applications has a unique Nash equilibrium, but we discuss examples in which this is not true.) In Section 1.3 we return to theory. We first define the no tion of a mixed strategy, which we will interpret in terms of one player's uncertainty about what another player will do. We then state and discuss Nash's (1950) Theorem, which guarantees that a Nash equilibrium (possibly involving mixed strategies) exists in a broad class of games. Since we present first basic theory in Sec tion 1.1, then applications in Section 1.2, and finally more theory in Section 1.3, it should be apparent that mastering the additional theory in Section 1.3 is not a prerequisite for understanding the applications in Section 1.2. On the other hand, the ideas of a mixed strategy and the existence of equilibrium do appear (occasionally) in later chapters. This and each subsequent chapter concludes with problems, suggestions for further reading, and references. 1.1 Basic Theory: Normal-Form Games and Nash Equilibrium 1.I.A Normal-Form Representation of Games In the normal-form representation of a game, each player simul taneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. We illustrate the normal-form representation with a classic example - The Prisoners' Dilemma. Two suspects are arrested and charged with a crime. The police lack sufficient evidence to convict the sus pects, unless at least one confesses. The police hold the suspects in