Game Theory: An Introduction

Cover
Title Page
Copyright
Contents
Preface for the Third Edition
Preface for the Second Edition
Preface for the First Edition
Acknowledgments
Introduction
Chapter 1 Matrix Two‐Person Games
	1.1 What Is Game Theory?
	1.2 Motivating Examples
		1.2.1 Three Card Poker
		1.2.2 Simplified Baseball
		1.2.3 2×2 NIM
	1.3 Mathematical Setup
		1.3.1 Definition of a Matrix Game
		1.3.2 Saddle Points: What It Means to be Optimal
	Problems
	1.4 Mixed Strategies
		1.4.1 Definition of Mixed Strategies
		1.4.2 Optimal Mixed Strategies
		1.4.3 Best Response Strategies
		1.4.4 Dominated Strategies
	Problems
	1.5 The Indifference Principle and Completely Mixed Games
		1.5.1 2×2 Games
		1.5.2 Completely Mixed Games and Invertible Matrix Games
		1.5.3 An Application: Optimal Target Choice and Defense
	Problems
	1.6 Finding Saddle Points in General
		1.6.1 Graphical Methods
		1.6.2 The n×m Case and Linear Programming
		1.6.2 Matrix Games and Linear Programming
			1.6.2.0 A Little Aside on Linear Programming
		1.6.3 Using Calculus
		1.6.4 Symmetric Games
	Problems
	1.7 Existence of Saddle Points: The Von Neumann Minimax Theorem
		1.7.1 Statement of the Minimax Theorem
		1.7.2 Von Neumann\'s Theorem Guarantees Matrix Games Have Saddle Points
	Problems
	1.8 Review Problems
	Problems
	1.9 Appendix: A Proof of the von Neumann Minimax Theorem
Chapter 2 Two‐Person Nonzero Sum Games
	2.1 The Basics
		2.1.1 Prisoner\'s Dilemma
	Problems
	2.2 2×2 Bimatrix Games, Best Response, Equality of Payoffs
	Problems
	2.3 Interior Mixed Nash Points by Calculus
		2.3.1 Calculus Method for Interior Nash
	Problems
		2.3.2 Existence of a Nash Equilibrium for Bimatrix Games
	2.4 Nonlinear Programming Method for Nonzero Sum Two‐Person Games
	Summary of Methods for Finding Mixed Nash Equilibria
	Problems
	2.5 Correlated Equilibria
		2.5.1 Motivating Example
		2.5.2 Definition of Correlated Equilibrium and Social Welfare
	Problems
	2.6 Choosing Among Several Nash Equilibria (Optional)
	Problems
	Bibliographic Notes
Chapter 3 Games in Extensive Form: Sequential Decision Making
	3.1 Introduction to Game Trees/Extensive form of Games
		3.1.1 Gambit
	Problems
	3.2 Backward Induction and Subgame Perfect Equilibrium
	Problems
		3.2.1 Subgame Perfect Equilibrium
		3.2.2 Examples of Extensive Games Using Gambit
	3.3 Behavior Strategies in Extensive Games
	Problems
	3.4 Extensive Games with Imperfect Information
		3.4.1 Bayesian Games and Bayesian Equilibria
			3.4.1.1 Separating and Pooling PBEs
	Problems
	Bibliographic Notes
Chapter 4 N‐Person Nonzero Sum Games and Games with a Continuum of Strategies
	4.1 Motivating Examples
	4.2 The Basics
		4.2.1 Do We Have Mixed Strategies in Continuous Games
		4.2.2 Existence of Pure NE
	Problems
	4.3 Economics Applications of Nash Equilibria
	Problems
	4.4 Duels
	Problems
	4.5 Auctions
		4.5.1 Complete Information
	Problems
		4.5.2 Symmetric Independent Private Value Auctions
	Problems
	4.6 Stable Matching, Marriage, and Residencies
	Problems
	4.7 Selected Chapter Problems
	Problems
	Bibliographic Notes
Chapter 5 Repeated Games
	5.1 Games Repeated Until ...
	5.2 Grim‐Trigger in General
		5.2.1 A Better Estimate for the Discount Factor
		5.2.2 Folk Theorems
	Problems
	Bibliographic Notes
Chapter 6 Cooperative Games
	6.1 What Is a Cooperative Game?
	6.2 Coalitions and Characteristic Functions
	Problems
		6.2.1 More on the Core and Least Core
	Problems
	6.3 The Nucleolus
		6.3.1 An Exact Nucleolus for Three Player Games
	Problems
	6.4 The Shapley Value
	Problems
	Bibliographic Notes
Chapter 7 Bargaining
	7.1 Introduction
	7.2 The Nash Model with Security Point
	7.3 Threats
		7.3.1 Finding the Threat Strategies
			7.3.1.1 Summary Approach for Bargaining with Threat Strategies
			7.3.1.2 Another Way to Derive the Threat Strategies Procedure
	7.4 The Kalai–Smorodinsky Bargaining Solution
	7.5 Sequential Bargaining
	Problems
	Bibliographic Notes
Chapter 8 Evolutionary Stable Strategies and Population Games
	8.1 Evolution
		8.1.1 Properties of an ESS
	Problems
	8.2 Population Games
	8.3 The Von Neumann Minimax Theorem from Replicator Dynamics
	Problems
	Bibliographic Notes
Appendix A The Essentials of Matrix Analysis
Appendix B The Essentials of Probability
Appendix C The Mathematica Commands
	C.1 The Upper and Lower Values of a Game
	C.2 The Value of an Invertible Matrix Game with Mixed Strategies
	C.3 Solving Matrix Games
	C.4 Interior Nash Points
	C.5 Lemke–Howson Algorithm for Nash Equilibrium
	C.6 Is the Core Empty?
	C.7 Find and Plot the Least Core
	C.8 Nucleolus Procedure and Shapley Value
	C.9 Mathematica Code for Three‐Person Nucleolus
	C.10 Plotting the Payoff Pairs
	C.11 Bargaining Solutions
	C.12 Mathematica for Replicator Dynamics
Appendix D Biographies
	D.1 John Von Neumann
	D.2 John Forbes Nash
Selected Problem Solutions
References
Index
EULA