This textbook presents the basics of game theory both on an undergraduate level and on a more advanced mathematical level. It is the second, revised version of the successful 2008 edition. The book covers most topics of interest in game theory, including cooperative game theory. Part I presents introductions to all these topics on a basic yet formally precise level. It includes chapters on repeated games, social choice theory, and selected topics such as bargaining theory, exchange economies, and matching. Part II goes deeper into noncooperative theory and treats the theory of zerosum games, refinements of Nash equilibrium in strategic as well as extensive form games, and evolutionary games. Part III covers basic concepts in the theory of transferable utility games, such as core and balancedness, Shapley value and variations, and nucleolus. Some mathematical tools on duality and convexity are collected in Part IV. Every chapter in the book contains a problem section. Hints, answers and solutions are included.
Author(s): Hans Peters
Series: Springer Texts in Business and Economics
Edition: 2nd ed. 2015
Publisher: Springer
Year: 2015
Language: English
Pages: 511
Preface
About the Second Edition
How to Use This Book
Reference
Contents
1 Introduction
1.1 A Definition
1.2 Some History
1.3 Examples
1.3.1 Zero-Sum Games
1.3.1.1 The Battle of the Bismarck Sea
1.3.1.2 Matching Pennies
1.3.2 Nonzero-Sum Games
1.3.2.1 Prisoners' Dilemma
1.3.2.2 Battle of the Sexes
1.3.2.3 Matching Pennies
1.3.2.4 A Cournot Game
1.3.3 Extensive Form Games
1.3.3.1 Sequential Battle of the Sexes
1.3.3.2 Sequential Cournot
1.3.3.3 Entry Deterrence
1.3.3.4 Entry Deterrence with Incomplete Information
1.3.4 Cooperative Games
1.3.4.1 Three Cooperating Cities
1.3.4.2 The Glove Game
1.3.4.3 A Permutation Game
1.3.4.4 A Voting Game
1.3.5 Bargaining Games
1.3.5.1 A Division Problem
1.4 Cooperative Versus Noncooperative Game Theory
1.5 Problems
1.6 Notes
References
Part I Thinking Strategically
2 Finite Two-Person Zero-Sum Games
2.1 Basic Definitions and Theory
2.2 Solving 2n Games and m2 Games
2.2.1 2n Games
2.2.2 m2 Games
2.2.3 Strict Domination
2.3 Problems
2.4 Notes
Reference
3 Finite Two-Person Games
3.1 Basic Definitions and Theory
3.2 Finding Nash Equilibria
3.2.1 Pure Nash Equilibria
3.2.2 22 Games
3.2.3 Strict Domination
3.3 Problems
3.4 Notes
References
4 Finite Extensive Form Games
4.1 The Extensive Form
4.2 The Strategic Form
4.3 Backward Induction and Subgame Perfection
4.4 Perfect Bayesian Equilibrium
4.5 Problems
4.6 Notes
References
5 Finite Games with Incomplete Information
5.1 Player Types
5.2 Static Games of Incomplete Information
5.2.1 Battle-of-the-Sexes with One-Sided Incomplete Information
5.2.2 Battle-of-the-Sexes with Two-Sided Incomplete Information
5.3 Signaling Games
5.3.1 An Example
5.3.2 Computing Perfect Bayesian Equilibria in the Extensive Form
5.3.3 The Intuitive Criterion
5.3.4 Another Example
5.4 Problems
5.5 Notes
References
6 Noncooperative Games: Extensions
6.1 General Framework: Strategic Games
6.2 Cournot Quantity Competition
6.2.1 Simple Version with Complete Information
6.2.1.1 Pareto Optimality
6.2.2 Simple Version with Incomplete Information
6.3 Bertrand Price Competition
6.4 Stackelberg Equilibrium
6.5 Auctions
6.5.1 Complete Information
6.5.2 Incomplete Information
6.5.3 Incomplete Information: A Double Auction
6.6 Mixed Strategies and Incomplete Information
6.7 Sequential Bargaining
6.7.1 Finite Horizon Bargaining
6.7.2 Infinite Horizon Bargaining
6.8 Problems
6.9 Notes
References
7 Repeated Games
7.1 Subgame Perfect Equilibrium
7.1.1 The Prisoners' Dilemma
7.1.2 Some General Observations
7.1.3 Another Example
7.2 Nash Equilibrium
7.2.1 An Example
7.2.2 A Folk Theorem for Nash Equilibrium
7.2.3 Another Example
7.3 Problems
7.4 Notes
References
8 An Introduction to Evolutionary Games
8.1 Symmetric Two-Player Games and Evolutionary Stable Strategies
8.2 Replicator Dynamics and Evolutionary Stability
8.3 Asymmetric Games
8.4 Problems
8.5 Notes
References
9 Cooperative Games with Transferable Utility
9.1 Examples and Preliminaries
9.2 The Core
9.3 The Shapley Value
9.4 The Nucleolus
9.5 Problems
9.6 Notes
References
10 Cooperative Game Models
10.1 Bargaining Problems
10.1.1 The Nash Bargaining Solution
10.1.2 Relation with the Rubinstein Bargaining Procedure
10.2 Exchange Economies
10.3 Matching Problems
10.4 House Exchange
10.5 Problems
10.6 Notes
References
11 Social Choice
11.1 Introduction and Preliminaries
11.1.1 An Example
11.1.2 Preliminaries
11.2 Arrow's Theorem
11.3 The Gibbard-Satterthwaite Theorem
11.4 Problems
11.5 Notes
References
Part II Noncooperative Games
12 Matrix Games
12.1 The Minimax Theorem
12.2 A Linear Programming Formulation
12.3 Problems
12.4 Notes
References
13 Finite Games
13.1 Existence of Nash Equilibrium
13.2 Bimatrix Games
13.2.1 Pure and Mixed Strategies in Nash Equilibrium
13.2.2 Extension of the Graphical Method
13.2.3 A Mathematical Programming Approach
13.2.4 Matrix Games
13.2.5 The Game of Chess: Zermelo's Theorem
13.3 Iterated Dominance and Best Reply
13.4 Perfect Equilibrium
13.5 Proper Equilibrium
13.6 Strictly Perfect Equilibrium
13.7 Correlated Equilibrium
13.8 A Characterization of Nash Equilibrium
13.9 Problems
13.10 Notes
References
14 Extensive Form Games
14.1 Extensive Form Structures and Games
14.2 Pure, Mixed and Behavioral Strategies
14.3 Nash Equilibrium and Refinements
14.3.1 Subgame Perfect Equilibrium
14.3.2 Perfect Bayesian and Sequential Equilibrium
14.4 Problems
14.5 Notes
References
15 Evolutionary Games
15.1 Symmetric Two-Player Games
15.1.1 Symmetric 22-Games
15.2 Evolutionary Stability
15.2.1 Evolutionary Stable Strategies
15.2.2 The Structure of the Set ESS(A)
15.2.3 Relations with Other Refinements
15.2.4 Other Characterizations of ESS
15.2.4.1 Uniform Invasion Barriers
15.2.4.2 Local Superiority
15.2.4.3 Local Strict Efficiency
15.3 Replicator Dynamics and ESS
15.3.1 Replicator Dynamics
15.3.2 Symmetric 22 Games
15.3.3 Dominated Strategies
15.3.4 Nash Equilibrium Strategies
15.3.5 Perfect Equilibrium Strategies
15.4 Problems
15.5 Notes
References
Part III Cooperative Games
16 TU-Games: Domination, Stable Sets, and the Core
16.1 Imputations and Domination
16.2 The Core and the Domination-Core
16.3 Simple Games
16.4 Stable Sets
16.5 Balanced Games and the Core
16.6 Problems
16.7 Notes
References
17 The Shapley Value
17.1 Definition and Shapley's Characterization
17.2 Other Characterizations
17.2.1 Dividends
17.2.2 Strong Monotonicity
17.2.3 Multilinear Extension
17.3 Potential and Reduced Game
17.3.1 The Potential Approach to the Shapley Value
17.3.2 Reduced Games
17.4 Problems
17.5 Notes
References
18 Core, Shapley Value, and Weber Set
18.1 The Weber Set
18.2 Convex Games
18.3 Random Order Values
18.4 Weighted Shapley Values
18.5 Problems
18.6 Notes
References
19 The Nucleolus
19.1 An Example
19.2 The Lexicographic Order
19.3 The (Pre-)Nucleolus
19.4 The Kohlberg Criterion
19.5 Computation of the Nucleolus
19.6 A Characterization of the Pre-nucleolus
19.7 Problems
19.8 Notes
References
20 Special Transferable Utility Games
20.1 Assignment and Permutation Games
20.2 Flow Games
20.3 Voting Games: The Banzhaf Value
20.4 Problems
20.5 Notes
References
21 Bargaining Problems
21.1 The Bargaining Problem
21.2 The Raiffa–Kalai–Smorodinsky Solution
21.3 The Egalitarian Solution
21.4 Noncooperative Bargaining
21.5 Games with Nontransferable Utility
21.6 Problems
21.7 Notes
References
Part IV Tools, Hints and Solutions
22 Tools
22.1 Some Definitions
22.2 A Separation Theorem
22.3 Lemmas of the Alternative
22.4 The Duality Theorem of Linear Programming
22.5 Some Fixed Point Theorems
22.6 The Birkhoff–von Neumann Theorem
22.7 The Max-Flow Min-Cut Theorem
22.8 Problems
22.9 Notes
References
23 Review Problems for Part I
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
References
24 Hints, Answers and Solutions
Problems of Chapter 1
Problems of Chapter 2
Problems of Chapter 3
Problems of Chapter 4
Problems of Chapter 5
Problems of Chapter 6
Problems of Chapter 7
Problems of Chapter 8
Problems of Chapter 9
Problems of Chapter 10
Problems of Chapter 11
Problems of Chapter 12
Problems of Chapter 13
Problems of Chapter 14
Problems of Chapter 15
Problems of Chapter 16
Problems of Chapter 17
Problems of Chapter 18
Problems of Chapter 19
Problems of Chapter 20
Problems of Chapter 21
Problems of Chapter 22
Problems of Chapter 23
Reference
References
Index
