Game Theory : An Introduction with Step-by-Step Examples

Preface
Organization of the Book
How to Use This Textbook
Ancillary Materials
Acknowledgments
References
Contents
List of Figures
List of Tables
1 Introduction to Games and Their Representation
1.1 Introduction
1.2 What Is Game Theory?
1.3 Main Elements in a Game
1.3.1 Players
1.3.2 Strategies
1.3.3 Payoffs
1.4 Two Graphical Approaches
1.4.1 Matrices
1.4.2 Game Trees
1.5 Introducing Imperfect Information in Game Trees
1.6 Identifying Equilibrium Behavior
1.6.1 Does an Equilibrium Exist?
1.6.2 Is the Equilibrium Unique?
1.6.3 Is the Equilibrium Robust to Small Payoff Changes?
1.6.4 Is the Equilibrium Pareto Optimal?
References
2 Equilibrium Dominance
2.1 Introduction
2.2 Strictly Dominated Strategies
2.3 Iterated Deletion of Strictly Dominated Strategies
2.3.1 Does the Order of Deletion Matter in IDSDS?
2.3.2 Deleting More Than One Strategy at a Time
2.3.3 Multiple Equilibrium Predictions
2.4 Applying IDSDS in Common Games
2.4.1 Prisoner’s Dilemma Game
2.4.2 Coordination Games—The Battle of the Sexes Game
2.4.3 Pareto Coordination Game—The Stag Hunt Game
2.4.4 Anticoordination Game—The Game of Chicken
2.4.5 Symmetric and Asymmetric Games
2.5 Allowing for Randomizations to Bring IDSDS Further
2.5.1 What If IDSDS Has No Bite?
2.6 Evaluating IDSDS as a Solution Concept
2.7 Weakly Dominated Strategies
2.7.1 Deletion Order Matters in IDWDS
2.7.2 IDSDS Vs. IDWDS
2.8 Strictly Dominant Strategies
2.8.1 Evaluating SDE as a Solution Concept
Exercises
Reference
3 Nash Equilibrium
3.1 Introduction
3.2 Best Response
3.2.1 Finding Best Responses with Discrete Strategy Spaces
3.2.2 Finding Best Responses with Continuous Strategy Spaces
3.3 Deleting Strategies That Are Never a Best Response
3.4 Rationalizability
3.4.1 Evaluating Rationalizability as a Solution Concept
3.5 Applications of Rationalizability
3.5.1 Finding NBRs in the Beauty Contest
3.5.2 Finding NBRs in the Cournot Duopoly
3.6 Nash Equilibrium
3.7 Finding Nash Equilibria in Common Games
3.7.1 Prisoner’s Dilemma Game
3.7.2 Coordination Game—The Battle of the Sexes Game
3.7.3 Pareto Coordination Game—The Stag Hunt Game
3.7.4 Anticoordination Game—The Game of Chicken
3.7.5 Multiple Nash Equilibria
3.8 Relationship Between NE and IDSDS
3.9 What If We Find No NEs?
3.10 Evaluating NE as a Solution Concept
Appendix: Equilibrium Selection
Pareto Dominance
Risk Dominance
Exercises
References
4 Nash Equilibria in Games with Continuous Action Spaces
4.1 Introduction
4.2 Quantity Competition
4.2.1 Quantity Competition with Homogeneous Goods and Two Firms
4.2.2 Extending Quantity Competition to Nge2 Firms
4.2.3 Quantity Competition with Heterogeneous Goods
4.3 Price Competition
4.3.1 Price Competition with Homogeneous Goods
4.3.2 Price Competition with Heterogeneous Goods
4.4 Public Good Game
4.4.1 Inefficient Equilibrium
4.5 Electoral Competition
4.5.1 Alternative Proof to the Electoral Competition game
Exercises
References
5 Mixed Strategy Nash Equilibrium
5.1 Introduction
5.2 Mixed Strategy
5.3 Mixed Strategy Nash Equilibrium
5.4 Finding Mixed Strategy Equilibria
5.4.1 Graphical Representation of Best Responses
5.5 Some Lessons
5.6 Extensions
5.6.1 Mixed Strategy Equilibria in Games with kge3 Pure Strategies
5.6.2 Finding Mixed Strategy Equilibria in Games with Nge2 Players
5.7 Strictly Competitive Games
5.7.1 Strictly Competitive Games
5.7.2 Zero-Sum Games
5.7.3 Security Strategies
5.8 Security Strategies and NE
5.9 Correlated Equilibrium
5.9.1 Public or Private Recommendations?
5.10 Equilibrium Refinements in Strategic-Form Games (Technical)
5.10.1 Trembling-Hand Perfect Equilibrium
5.10.2 Proper Equilibrium
Appendix—NE Existence Theorem (Technical)
Fixed-Point Theorems, an Introduction
Nash Existence Theorem
Exercises
References
6 Subgame Perfect Equilibrium
6.1 Introduction
6.2 Tree rules
6.2.1 Actions vs. Strategies
6.3 Why Don’t We Just Find the Nash Equilibrium of the Game Tree?
6.4 Subgames
6.4.1 What If the Game Tree Has Information Sets?
6.5 Subgame Perfect Equilibrium
6.5.1 Finding SPEs in Games Without Information Sets
6.5.2 Finding SPEs in Game Trees with Information Sets
6.6 Evaluating SPE as a Solution Concept
6.7 Applications
6.7.1 Stackelberg Game of Sequential Quantity Competition
6.7.2 Sequential Public Good Game
6.7.3 Ultimatum Bargaining Game
6.7.4 Two-Period Alternating-Offers Bargaining Game
6.7.5 Some Tricks About Solving Alternating-Offer Bargaining Games
6.7.6 Alternating-Offer Bargaining Game with Infinite Periods
Appendix—Mixed and Behavioral Strategies
Exercises
References
7 Repeated Games
7.1 Introduction
7.2 Repeating the Game Twice
7.3 Repeating the Game Tge2 Times
7.4 Repeating the Game Infinitely Many Times
7.4.1 Uncooperative Outcome
7.4.2 Cooperative Outcome
7.4.3 Cooperative Outcome—Extensions
7.5 Folk Theorem
7.5.1 Feasible and Individually Rational Payoffs
7.5.2 Folk Theorem and Cooperation
7.6 Application to Collusion in Oligopoly
7.6.1 Minimal Discount Factor Supporting Collusion
7.6.2 Other Collusive GTS
7.7 What if the Stage Game has More than One NE?
7.8 Modified GTSs
7.8.1 An Eye for an Eye
7.8.2 Short and Nasty Punishments
7.8.3 Imperfect Monitoring
Exercises
References
8 Bayesian Nash Equilibrium
8.1 Introduction
8.2 Background
8.2.1 Players’ Types and Their Associated Probability
8.2.2 Strategies Under Incomplete Information
8.2.3 Representing Asymmetric Information as Incomplete Information
8.2.4 Best Response Under Incomplete Information
8.3 Bayesian Nash Equilibrium
8.3.1 Ex-ante and Ex-post Stability
8.4 Finding BNEs—First Approach: Build the Bayesian Normal Form
8.5 Finding BNEs—Second Approach: Focus on the Informed Player First
8.6 Evaluating BNE as a Solution Concept
8.7 What If Both Players Are Privately Informed?
Exercises
Reference
9 Auction Theory
9.1 Introduction
9.2 Auctions as Allocation Mechanisms
9.3 Second-price Auctions
9.3.1 Case 1: Bid Equal To Her Valuation
9.3.2 Case 2: Downward Deviations, Bidding Below Her Valuation
9.3.3 Case 3: Upward Deviations, Bidding Above Her Valuation
9.3.4 Discussion
9.4 First-Price Auctions
9.5 Efficiency in Auctions
9.6 Seller’s Expected Revenue
9.6.1 Expected Revenue in the FPA
9.6.2 Expected Revenue in the SPA
9.6.3 Revenue Equivalence Principle
9.7 Common-Value Auctions and the Winner’s Curse
9.7.1 Bid Shading Is a Must!
9.7.2 Equilibrium Bidding in Common-Value Auctions
Exercises
10 Perfect Bayesian Equilibrium
10.1 Introduction
10.2 Sequential-Move Games of Incomplete Information—Notation
10.3 BNE Prescribing Sequentially Irrational Behavior
10.4 Perfect Bayesian Equilibrium—Definition
10.5 A Tool to Find PBEs in Signaling Games
10.6 Finding PBEs in Games with one Information Set
10.6.1 Separating Strategy Profile (OB,NF)
10.6.2 Pooling Strategy Profile (OB,OF)
10.7 Finding PBEs in Games with Two Information Sets
10.7.1 Separating Strategy Profile (EH,NEL)
10.7.2 Pooling Strategy Profile (NEH,NEL)
10.7.3 Insensible Off-the-Equilibrium Beliefs
10.8 Evaluating PBE as a Solution Concept
10.9 Semi-Separating PBE
10.10 Extensions
10.10.1 What if the Receiver has More than Two Available Responses?
10.10.2 What if the Sender has More than Two Available Messages?
10.10.3 What if the Sender has More than Two Types?
10.10.4 Other Extensions
Exercises
Reference
11 Equilibrium Refinements
11.1 Introduction
11.2 Intuitive Criterion
11.2.1 A Six-Step Tool to Apply the Intuitive Criterion
11.2.2 Separating Equilibria Survive the Intuitive Criterion
11.3 D1 Criterion
11.3.1 Applying the D1 Criterion—An Example
11.3.2 Discrete and Continuous Responses
11.3.3 Comparing Intuitive and Divinity Criteria
11.3.4 Other Refinement Criteria
11.4 Sequential Equilibrium
11.4.1 Finding Sequential Equilibria
11.4.2 Separating PBEs that Are Also SEs
11.4.3 A Pooling PBE that Is Not a SE
11.4.4 A Pooling PBE that Is Also A SE
Exercises
References
12 Signaling Games with Continuous Messages
12.1 Introduction
12.2 Utility Functions with Continuous Actions
12.3 Complete Information
12.4 Separating PBE
12.4.1 Separating PBE—Applying the Intuitive Criterion
12.4.2 Separating PBE—Applying the D1 Criterion
12.5 Pooling PBE
12.5.1 Other Pooling PBEs
12.5.2 Pooling PBE—Applying the Intuitive Criterion
12.6 Can Signaling Be Welfare Improving?
12.7 What If the Sender Has Three Types?
12.7.1 Separating PBEs
12.7.2 Separating PBE—Applying the Intuitive Criterion
12.7.3 Separating PBE—Applying the D1 Criterion
Appendix: Equilibrium Refinements
Intuitive Criterion
D1 Criterion
Exercises
References
13 Cheap Talk Games
13.1 Introduction
13.2 Cheap Talk with Discrete Messages and Responses
13.2.1 Separating PBE
13.2.2 Pooling PBEs
13.3 Cheap Talk with Discrete Messages But Continuous Responses
13.3.1 Separating PBE
13.3.2 Pooling PBEs
13.4 Cheap Talk with Continuous Messages and Responses
13.4.1 Separating PBE
13.4.2 Equilibrium Number of Partitions
13.4.3 Interval Lengths in Equilibrium
13.5 Extensions
Exercises
References
Mathematical Appendix
A.1 Sets
A.2 Sequences
A.3 Functions
A.4 Limits and Continuity
A.5 Quadratic Equation
A.6 Cramer’s Rule
A.7 Differentiation
A.7.1 First- and Second-order Conditions
A.7.2 Inverse Function Theorem
A.8 Integration
References
Index