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 1Draft March 30, 2018, copyright by George J. Mailath. To be published by World Scientific Press. Contents Contents i Preface vii 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 . 11 1.3.1 The Reduced Normal Form . 15 1.4 Problems . 17 2 A First Look at Equilibrium 21 2.1 Nash Equilibrium . 21 2.1.1 Why Study Nash Equilibrium? . 25 2.2 Credible Threats and Backward Induction . 26 2.2.1 Backward Induction and Iterated Weak Domi- nance . 30 2.3 Subgame Perfection . 31 2.4 Mixing . 36 2.4.1 Mixed Strategies and Security Levels . 36 2.4.2 Domination and Optimality . 38 2.4.3 Equilibrium in Mixed Strategies . 43 2.4.4 Behavior Strategies . 48 2.5 Dealing with Multiplicity . 50 2.5.1 Refinements . 50 2.5.2 Selection . 52 2.6 Problems . 54 i 3 Games with Nature 65 3.1 An Introductory Example . 65 3.2 Purification . 67 3.3 Auctions and Related Games . 70 3.4 Games of Incomplete Information . 87 3.5 Higher Order Beliefs and Global Games . 93 3.6 Problems . 101 4 Nash Equilibrium 107 4.1 Existence of Nash Equilibria . 107 4.2 Foundations for Nash Equilibrium . 111 4.2.1 Boundedly Rational Learning . 111 4.2.2 Social Learning (Evolution) . 112 4.2.3 Individual learning . 120 4.3 Problems . 121 5 Refinements in Dynamic Games 129 5.1 Sequential Rationality . 129 5.2 Perfect Bayesian Equilibrium . 136 5.3 Sequential Equilibrium . 142 5.4 Problems . 146 6 Signaling 151 6.1 General Theory . 151 6.2 Job Market Signaling . 154 6.2.1 Full Information . 155 6.2.2 Incomplete Information . 156 6.2.3 Refining to Separation . 161 6.2.4 Continuum of Types . 162 6.3 Problems . 164 7 Repeated Games 173 7.1 Basic Structure . 173 7.1.1 The Stage Game . 173 7.1.2 The Repeated Game . 173 7.1.3 Subgame Perfection . 177 7.1.4 Automata . 178 7.1.5 Renegotiation-Proof Equilibria . 188 7.2 Modeling Competitive Agents . 189 7.3 Applications . 196 7.3.1 Efficiency Wages I . 196 7.3.2 Collusion Under Demand Uncertainty . 200 7.4 Enforceability, Decomposability, and A Folk Theorem . 203 7.5 Imperfect Public Monitoring . 209 7.5.1 Efficiency Wages II . 209 7.5.2 Basic Structure and Public Perfect Equilibria . 212 7.5.3 Automata . 213 7.6 Problems . 218 8 Topics in Dynamic Games 229 8.1 Dynamic Games and Markov Perfect Equilibria . 229 8.2 Coase Conjecture . 235 8.2.1 One and Two Period Example . 235 8.2.2 Infinite Horizon . 238 8.3 Reputations . 241 8.3.1 Two Periods . 241 8.3.2 Infinite Horizon . 244 8.3.3 Infinite Horizon with Behavioral Types . 247 8.4 Problems . 250 9 Bargaining 257 9.1 Axiomatic Nash Bargaining . 257 9.1.1 The Axioms . 257 9.1.2 Nash’s Theorem . 258 9.2 Rubinstein Bargaining . 260 9.2.1 The Stationary Equilibrium . 261 9.2.2 All Equilibria . 262 9.2.3 Impatience . 263 9.3 Outside Options . 264 9.3.1 Version I . 264 9.3.2 Version II . 267 9.4 Exogenous Risk of Breakdown . 270 9.5 Problems . 271 10 Introduction to Mechanism Design 277 10.1 A Simple Screening Example . 277 10.2 A Less Simple Screening Example . 281 10.3 The Take-It-or-Leave-It Mechanism . 286 10.4 Implementation . 288 10.5 Problems . 290 11 Dominant Strategy Mechanism Design 293 11.1 Social Choice and Arrow’s Impossibility Theorem . 293 11.2 Gibbard-Satterthwaite Theorem . 297 11.3 Efficiency in Quasilinear Environments . 302 11.4 Problems . 304 12 Bayesian Mechanism Design 307 12.1 The (Bayesian) Revelation Principle . 307 12.2 Efficiency in Quasilinear Environments . 309 12.3 Incomplete Information Bargaining . 312 12.3.1 The Impossibility of Ex Post Efficient Trade . 313 12.3.2 Maximizing Ex Ante Gains From Trade . 317 12.4 Auctions . 324 12.5 Problems . 328 13 Principal Agency 335 13.1 Introduction . 335 13.2 Observable Effort . 336 13.3 Unobservable Effort . 338 13.4 Unobserved Cost of Effort . 344 13.5 A Hybrid Model . 345 13.6 Problems . 349 14 Appendices 351 14.1 Proof of Theorem 2.4.1 . 351 14.2 Trembling Hand Perfection . 354 14.2.1 Existence and Characterization . 354 14.2.2 Extensive form trembling hand perfection . 356 14.3 Completion of the Proof of Theorem 12.3.1 . 358 14.4 Problems . 359 References 361 Index 371 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, reflecting my own idiosyncracies. 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 Chapter 7 (Repeated Games), and never both Sections 8.2 (Coase conjecture) and 8.3 (reputa- tions). Thanks to the many generations of Penn graduate students who were subjected to early versions of these notes, and made many helpful comments. vii Chapter 1 Normal and Extensive Form Games 1.1 Normal Form Games Most introductions to game theory start with the prisoners’ dilemma. Two suspects (I and II) are separately interrogated, after being ar- rested with incriminating evidence. The results of the interrogation are illustrated in Figure 1.1.1.1 Clearly, no matter what the other suspect does, it 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. Total output is shared equally. The result is given in Figure 1.1.2. With this formulation, no matter what the other partner does (E or S), the partner maximizes his/her payoff by shirking. The scenarios illustrated in Figures 1.1.1 and 1.1.2 are examples of normal form games. Definition 1.1.1. An n-player normal (or strategic) form game G is an n-tuple (S1;U1), : : : ; (S ;U ) , where for each i, f n n g 1Poundstone (1993) discusses the prisoners’ dilemma and its role in the early history of game theory in the context of the cold war. Leonard (2010) gives a fascinating history of the birth of game theory. 1 2 Chapter 1. Normal and Extensive Form Games II Confess Don’t confess I Confess 6; 6 0; 9 − − − Don’t confess 9; 0 1; 1 − − − Figure 1.1.1: The prisoners’ dilemma, with the numbers describing length of sentence. In each cell, the first number is player I’s sen- tence, while the second is player II’s. ES E 2; 2 1; 3 − S 3; 1 0; 0 − Figure 1.1.2: The prisoners’ dilemma, as a partnership game. In each cell, the first number is the row player’s payoff, while the second number is the column player’s payoff. • Si is a nonempty set, i’s strategy space, with typical element si, and n • Ui : k 1 Sk R, i’s payoff function. = ! Q We sometimes treat the payoff function as a vector-valued func- n n tion U : i 1 Si R . = ! Q n Notation: S : k 1 Sk; = = s : (s1; : : : ; s ) S; = n Q2 s i : (s1; : : : ; si 1; si 1; : : : ; sn) S i : k≠i Sk: − = − + 2 − = (s0; s i) : (s1; : : : ; si 1; s0; si 1; : : : ; sn) S: i − = − i + 2Q Example 1.1.1 (Sealed bid second price auction). Two bidders si- multaneously submit bids (in sealed envelopes) for an object. A bid is a nonnegative number, with i’s bid denoted bi R . Bidder 2 + 1.1. Normal Form Games 3 i’s value for the object (reservation price, willingness to pay) is de- noted vi. Object is awarded to the highest bidder, who pays the second highest bid. Resolving any tie with a fair coin toss, and taking expected val- ues, n 2;Si R ; and = = + vi bj; if bi > bj; 1 − Ui(b1; b2) 8 2 (vi bj), if bi bj; = > − = <0; if bi < bj: « > :> Example 1.1.2 (Sealed bid first price auction).