Department of Economics, University of California , Davis, CA 95616-8578

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This textbook provides an introduction to non-cooperative game theory. The printed version is divided into two volummes: Volume 1 covers the basic concepts, while Volume 2 is devoted to advanced topics. Volume 1 is divided into two parts: Part I deals with games with ordinal payoffs, while Part II covers games with cardinal payoffs. In each part we discuss both strategic-form games and dynamic games. Volume 2 is divided into three parts. The first part deals with the notions of knowledge, belief and common knowledge. The second part covers solution concepts for dynamic games and the third part develops the theory of games of incomplete information. The book is suitable for both self-study and an undergraduate or first-year graduate-level course in game theory. It is written to be accessible to anybody with high-school level knowledge of mathematics. At the end of each chapter there is a collection of exercises accompanied by detailed answers. There is a total of more than 180 exercises. The book is richly illustrated with approximately 400 figures.

Table of contents:

Game Theory

Giacomo Bonanno

Preface.................................................................................................................. 3 Contents ................................................................................................................5 1. Introduction...................................................................................................... 11 PART I: Games with ordinal payoffs......................................................................15 2. Ordinal games in strategic form..........................................................................17 2.1 Game frames and games 2.2 Strict and weak dominance 2.3 Second-price auction 1.4 The pivotal mechanism 2.5 Iterated deletion procedures 2.6 Nash equilibrium 2.7 Games with infinite strategy sets 2.8 Proofs of theorems 2.9 Exercises [23 exercises] 2.10 Solutions to exercises 3. Perfect information games...............................................................................75 3.1 Trees, frames and games 3.2 Backward induction 3.3 Strategies in perfect-information games 3.4 Relationship between backward induction and other solutions 3.5 Perfect-information games with two players 3.6 Exercises [13 exercises] 3.7 Solutions to exercises 4. General dynamic games.................................................................................117 4.1 Imperfect information 4.2 Strategies 4.3 Subgames 4.4 Subgame-perfect equilibrium 4.5 Games with chance moves 4.6 Exercises [15 exercises] 4.7 Solutions to exercises PART II: Games with cardinal payoffs............................................................. 167 5. Expected Utility.......................................................................................... 169 5.1 Money lotteries and attitudes to risk 5.2 Expected utility: theorems 5.3 Expected utility: the axioms 5.4 Exercises [14 exercises] 5.5 Solutions to exercises 6. Strategic-form games.................................................................................. 193 6.1 Strategic-form games with cardinal payoffs 6.2 Mixed strategies 6.3 Computing the mixed-strategy Nash equilibria 6.4 Strict dominance and rationalizability 6.5 Exercises [15 exercises] 6.6 Solutions to exercises 7. Extensive-form games.................................................................................. 227 7.1 Behavioral strategies in dynamic games 7.2 Subgame-perfect equilibrium revisited 7.3 Problems with subgame-perfect equilibrium 7.4 Exercises [9 exercises] 7.5 Solutions to exercises PART III: Knowledge, common knowledge, belief............................................ 259 8. Common knowledge.................................................................................... 261 8.1 Individual knowledge 8.2 Interactive knowledge 8.3 Common Knowledge 8.4 Exercises [14 exercises] 8.5 Solutions to exercises 9. Adding beliefs to knowledge........................................................................ 295 9.1 Sets and probability: brief review 9.2 Probabilistic beliefs 9.3 Conditional probability and Bayes’ rule 9.4 Changing beliefs in response to information 9.5 Harsanyi consistency of beliefs or like-mindedness 9.6 Agreeing to disagree 9.7 Proof of the Agreement Theorem 9.8 Exercises [28 exercises] 9.9 Solutions to exercises 10. Common knowledge of rationality.................................................................347 10.1 Models of strategic-form games 10.2 Common knowledge of rationality in strategic-form games 10.3 Common knowledge of rationality in extensive-form games 10.4 Proofs of theorems Appendix 9.E: Exercises [7 exercises] Appendix 9.S: Solutions to exercises









PART IV: Refinements of subgame-perfect equilibrium..................................... 367

11. Weak Sequential Equilibrium...................................................................... 369

11.1 Assessments and sequential rationality

11.2 Bayesian updating at reached information sets

10.3 A first attempt: Weak sequential equilibrium

10.4 Exercises [8 exercises]

10.5 Solutions to exercises.

12. Sequential Equilibrium................................................................................ 403

12.1 Consistent assessments

12.2 Sequential equilibrium

12.3 Is ‘consistency’ a good notion?

12.4 Exercises [6 exercises]

12.5 Solutions to exercises

13. Perfect Bayesian Equilibrium....................................................................... 429

13.1 Belief revision and AGM consistency

13.2 Bayesian consistency

13.3 Perfect Bayesian equilibrium.

13.4 Adding independence

13.5 Characterization of SE in temrs of PBE

13.6 History-based definition of extensive-form game

13.7 Proofs

13.8 Exercises [13 exercises]

13.9 Solutions to exercises

PART V: Incomplete Information...................................................................481

14. Static Games...........................................................................................483

14.1 Interactive situations with incomplete information

14.2 One-sided incomplete information

14.3 Two-sided incomplete information

14.4 Multi-sided incomplete information

14.5 Exercises [8 exercises]

14.6 Solutions to exercises

15. Dynamic Games........................................................................................521

15.1 One-sided incomplete information

15.2 Multi-sided incomplete information

15.3 Exercises [7 exercises]

15.4 Solutions to exercises

16. The type-space approach..................................... 567

16.1 Types of players

16.2 Types that know their own payoffs

16.3 The general case

16.4 Exercises [4 exercises]

16.5 Solutions to exercises

References....................................................................................................... 585