Game Theory An Introduction Answers

Nov 06, 2000  Contents vii 6 Extensive Games with Perfect Information: Illustrations 179 6.1 Introduction 179 6.2 The ultimatum game and the holdup game 179 Experimentsontheultimatumgame 181 6.3 Stackelberg’s model of duopoly 184.

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.

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 andother 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

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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

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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. DynamicGames............................................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

Game Theory might be better described as Strategy Theory, or Theory of Interactive Decision Making. A strategic situation involves two or more interacting players who make decisions while trying to anticipate the actions and reactions by others. Game theory studies the general principles that explain how people and organizations act in strategic situations.

Game theory studies strategy mainly through the analysis of different 'games'. A 'game' in game theory is a fully explicit structure which characterizes each player's set of actions, payoffs and possible outcomes under given rules of playing. Given this conditions, rational players act in such a way, that they maximize the expected value of their von Neumann-Morgenstern Utility. Games provide a simplified world within which to study strategy (as opposed to the real world where complexities get in the way of developing general principles).

Table of Contents

1 Introduction to Game Theory

Part I - Games with Perfect Information

2 Nash Equilibrium

1. Example: Prisoner's Dilemma
2. Example: Battle of the sexes (a.k.a. Bach or Stravinsky?)
3. Best response functions
4. Cournot's model of oligopoly
5. Bertrand's model of oligopoly
6. Auctions
7. Questions
8. Sources

3 Mixed Strategy Equilibrium

1. Randomization
2. Mixed strategy Nash equilibrium
3. Dominated actions
4. Example: expert diagnosis
5. Formation of beliefs

4 Extensive Games with Perfect Equilibrium

1. Strategies and outcomes
2. Nash equilibrium
3. Subgame perfect equilibrium
4. Stackelberg's model of duopoly
5. Adding simultaneous moves
6. Adding uncertainty

5 Coalitional Games

Part II - Games with Imperfect Information

6 Bayesian Games

1. Motivational Examples
2. Cournot's duopoly with imperfect information

7 Extensive Games with Imperfect Information

1. Strategies
2. Nash equilibrium
3. Beliefs
4. Signaling games

Part III - Real World Examples

8 TV Game Shows

9 Politics

See Also

Wikipedia Articles on Game Theory Related Topics

• Prisoner's Dilemma.
• John von Neumann and John Forbes Nash.
• Game semantics, an approach to establish the notion of truth in mathematical logic, in another way than Tarski did, using game theoretical concepts