Modeling Strategic Behavior1 A Graduate Introduction to Game Theory and Mechanism Design George J. Mailath Department of Economics, University of Pennsylvania Research School of Economics, Australian National University 1September 16, 2020, copyright by George J. Mailath. World Scientific Press published the November 02, 2018 version. Any significant correc- tions or changes are listed at the back. To Loretta Preface These notes are based on my lecture notes for Economics 703, a first-year graduate course that I have been teaching at the Eco- nomics Department, University of Pennsylvania, for many years. It is impossible to understand modern economics without knowl- edge of the basic tools of game theory and mechanism design. My goal in the course (and this book) is to teach those basic tools so that students can understand and appreciate the corpus of modern economic thought, and so contribute to it. A key theme in the course is the interplay between the formal development of the tools and their use in applications. At the same time, extensions of the results that are beyond the course, but im- portant for context are (briefly) discussed. While I provide more background verbally on many of the exam- ples, I assume that students have seen some undergraduate game theory (such as covered in Osborne, 2004, Tadelis, 2013, and Wat- son, 2013). In addition, some exposure to intermediate microeco- nomics and decision making under uncertainty is helpful. Since these are lecture notes for an introductory course, I have not tried to attribute every result or model described. The result is a somewhat random pattern of citations and references. There is much more here than can be covered in a one semester course. I often do not cover Section 4.2 (Foundations of Nash Equi- librium), selectively cover material in Section 3.3 and Chapter 7 (Re- peated Games), and never both Sections 8.2 (Coase conjecture) and 8.3 (Reputations). I have been very lucky in my coauthors and colleagues (par- ticularly Larry Samuelson and Andrew Postlewaite), from whom I learned a tremendous amount. Thanks to the many generations iii iv Preface of Penn graduate students who were subjected to early versions of these notes, and made many helpful comments. Thanks also to Larry Samuelson and Tilman Börgers for helpful comments. Fi- nally, thanks to Ashwin Kambhampati and Changhwa Lee, who did an outstanding job proofreading these notes. Game Theory in Economics Game theory studies the strategic interactions of agents (often called “players” or decision-makers). An example of a strategic interaction is the pricing behavior of two petrol (gas) stations on the same in- tersection. Each station, in choosing its price, will both respond to the current price of the other station and to how it believes the other station will respond to its price. To study strategic interactions, it is useful to use parlor games such as chess and poker as examples. The first game theorists were mathematicians, and viewed the study of strategic interactions (be- ginning with parlor games) as applied mathematics.1 The goal was to calculate a solution, which was a prediction: How would rational players behave? A solution can also be a recommendation: How should a rational player behave (assuming the other player is ratio- nal)? The perspective of these notes is that of an economist. Economists are social scientists and, as such, want to understand social behav- ior. Any model simplifies the situation being modeled, and models of strategic interactions in an economic or social context are no different. The resulting modeling choices make the use of game theory within economics very different from game theory as ap- plied mathematics (which takes the rules of the game as given). In particular, the modeling choices reflect the modeler’s judgment as to what the players treat as strategically relevant. This judgment determines both the choices of strategy spaces (the actions and in- formation players have), and the choice of solution concept. There is often a subtle interplay between the question under investiga- tion, the modeling choices (including that of the solution concept), 1Leonard (2010) gives a fascinating history of the birth of game theory. Preface v and the resulting analysis. vi Preface Contents Preface iii Game Theory in Economics. iv 1 Normal and Extensive Form Games 1 1.1 Normal Form Games . .1 1.2 Iterated Deletion of Dominated Strategies . .9 1.3 Extensive Form Games . 12 1.3.1 The Reduced Normal Form . 16 1.4 Problems . 18 2 A First Look at Equilibrium 21 2.1 Nash Equilibrium . 21 2.1.1 Why Study Nash Equilibrium? . 26 2.2 Credible Threats and Backward Induction . 27 2.2.1 Backward Induction and Iterated Weak Domi- nance . 31 2.3 Subgame Perfection . 33 2.4 Mixing . 38 2.4.1 Mixed Strategies and Security Levels . 38 2.4.2 Domination and Optimality . 40 2.4.3 Equilibrium in Mixed Strategies . 45 2.4.4 Behavior Strategies . 50 2.5 Dealing with Multiplicity . 52 2.5.1 Refinements . 52 2.5.2 Selection . 56 2.6 Problems . 58 vii viii Contents 3 Games with Nature 69 3.1 An Introductory Example . 69 3.2 Purification . 71 3.3 Auctions and Related Games . 75 3.4 Games of Incomplete Information . 91 3.5 Higher Order Beliefs and Global Games . 97 3.6 Problems . 106 4 Nash Equilibrium: Existence and Foundations 113 4.1 Existence . 113 4.2 Learning/Evolutionary Foundations . 117 4.2.1 Social Learning (Evolutionary Game Theory) . 119 4.2.2 Individual learning . 128 4.3 Problems . 129 5 Nash Equilibrium Refinements in Dynamic Games 139 5.1 Sequential Rationality . 139 5.2 Perfect Bayesian Equilibrium . 147 5.3 Sequential Equilibrium . 152 5.4 Problems . 158 6 Signaling 165 6.1 General Theory . 165 6.2 Job Market Signaling . 169 6.2.1 Full Information . 170 6.2.2 Incomplete Information . 171 6.2.3 Refining to Separation . 176 6.2.4 Continuum of Types . 178 6.3 Problems . 180 7 Repeated Games 191 7.1 Perfect Monitoring . 191 7.1.1 The Stage Game . 191 7.1.2 The Repeated Game . 191 7.1.3 Subgame Perfection . 195 7.1.4 Automata . 198 7.1.5 Renegotiation-Proof Equilibria . 208 7.2 Short-Lived Players and Modeling Competitive Agents . 209 Contents ix 7.3 Applications . 216 7.3.1 Efficiency Wages I . 216 7.3.2 Collusion Under Demand Uncertainty . 220 7.4 Enforceability, Decomposability, and a Folk Theorem . 223 7.5 Imperfect Public Monitoring . 230 7.5.1 Efficiency Wages II . 230 7.5.2 Public Perfect Equilibria . 232 7.5.3 Automata . 234 7.6 Problems . 239 8 Topics in Dynamic Games 251 8.1 Dynamic Games and Markov Perfect Equilibria . 251 8.2 Disappearance of Monopoly Power and the Coase Conjecture . 258 8.2.1 One and Two Period Example . 258 8.2.2 Infinite Horizon . 261 8.3 Reputations . 264 8.3.1 Two Periods . 264 8.3.2 Infinite Horizon . 267 8.3.3 Infinite Horizon with Behavioral Types . 270 8.4 Problems . 273 9 Bargaining 281 9.1 Axiomatic Nash Bargaining . 281 9.1.1 The Axioms . 281 9.1.2 Nash’s Theorem . 282 9.2 Rubinstein (1982) Bargaining . 283 9.2.1 The Stationary Equilibrium . 284 9.2.2 All Equilibria . 286 9.2.3 Impatience . 288 9.3 Outside Options . 289 9.3.1 Version I . 289 9.3.2 Version II . 292 9.4 Exogenous Risk of Breakdown . 295 9.5 Problems . 296 x Contents 10 Introduction to Mechanism Design 303 10.1 A Simple Screening Example . 303 10.2 A Less Simple Screening Example . 307 10.3 The Take-It-or-Leave-It Mechanism . 313 10.4 Implementation . 314 10.5 Problems . 318 11 Dominant Strategy Mechanism Design 321 11.1 Social Choice and Arrow’s Impossibility Theorem . 321 11.2 Dominant Strategy Implementation and the Gibbard-Satterthwaite Theorem . 325 11.3 Efficiency in Quasilinear Environments . 330 11.4 Problems . 333 12 Bayesian Mechanism Design 335 12.1 The Bayesian Revelation Principle . 335 12.2 Efficiency in Quasilinear Environments . 337 12.3 Incomplete Information Bargaining . 340 12.3.1 The Impossibility of Ex Post Efficient Trade . 341 12.3.2 Maximizing Ex Ante Gains From Trade . 345 12.4 Independent Private Values Auctions . 352 12.5 Problems . 357 13 Principal Agency 363 13.1 Introduction . 363 13.2 Observable Effort . 364 13.3 Unobservable Effort (Moral Hazard) . 366 13.4 Unobserved Cost of Effort (Adverse Selection) . 372 13.5 A Hybrid Model . 373 13.6 Problems . 377 14 Appendices 379 14.1 Proof of Theorem 2.4.1 . 379 14.2 Trembling Hand Perfection . 382 14.2.1 Existence and Characterization . 382 14.2.2 Extensive Form Trembling Hand Perfection . 384 14.3 Completion of Proof of Theorem 12.3.1 . 386 14.4 Problems . 387 Contents xi References 389 Index 401 Corrections to the 2019 Published version 407 xii Contents Chapter 1 Normal and Extensive Form Games 1.1 Normal Form Games Most introductions to game theory start with the prisoner’s dilemma.1 Two suspects (I and II) are separately interrogated. The prosecu- tors have sufficient evidence to convict each of a minor offence, but wish to convict them of a major offence. The potential results of the interrogation are illustrated in Figure 1.1.1. Clearly, no matter what the other suspect does, it is always better to confess than not confess. This game is often interpreted as a partnership game, in which two partners simultaneously choose between exerting effort and shirking. Effort E produces an output of 6 at a cost of 4, while shirk- ing S yields no output at no cost.