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