Understanding Game Theory: Introduction To The Analysis Of Many Agent Systems With Competition And Cooperation

by Vasily N Kolokoltsov, Oleg A Malafeyev

Overview

Steadily growing applications of game theory in modern science (including psychology, biology and economics) require sources to provide rapid access in both classical tools and recent developments to readers with diverse backgrounds. This book on game theory, its applications and mathematical methods, is written with this objective in mind.The book gives a concise but wide-ranging introduction to games including older (pre-game theory) party games and more recent topics like elections and evolutionary games and is generously spiced with excursions into philosophy, history, literature and politics. A distinguished feature is the clear separation of the text into two parts: elementary and advanced, which makes the book ideal for study at various levels.Part I displays basic ideas using no more than four arithmetic operations and requiring from the reader only some inclination to logical thinking. It can be used in a university degree course without any (or minimal) prerequisite in mathematics (say, in economics, business, systems biology), as well as for self-study by school teachers, social and natural scientists, businessmen or laymen.Part II is a rapid introduction to the mathematical methods of game theory, suitable for a mathematics degree course of various levels. It includes an advanced material not yet reflected in standard textbooks, providing links with the exciting modern developments in financial mathematics (rainbow option pricing), tropical mathematics, statistical physics (interacting particles) and discusses structural stability, multi-criteria differential games and turnpikes.To stimulate the mathematical and scientific imagination, graphics by a world-renowned mathematician and mathematics imaging artist, A T Fomenko, are used. The carefully selected works of this artist fit remarkably into the many ideas expressed in the book.

Product Details

ISBN-13:	9789814291712
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/21/2010
Pages:	300
Product dimensions:	6.00(w) x 9.00(h) x 0.80(d)

Table of Contents

Preface vii

Basic ideas 1

1 Around the prisoner's dilemma 3

1.1 What is a two-player game? 3

1.2 Prisoner's dilemma. Dominated strategies and Pareto optimality 7

1.3 Prisoner's dilemma for crooks, warriors and opera lovers 9

1.4 Discrete duopoly models and common pool resources; public goods 12

1.5 Common knowledge, rationality and iterated elimination of strictly dominated strategies 13

1.6 Weak dominance; debtors and creditors 15

1.7 Nash equilibrium 16

1.8 Battle of the sexes and Kant's categorical imperative 19

1.9 Chicken game and the Cuban missile crisis 21

1.10 Social dilemmas 22

1.11 Guaranteed payoff, minimax strategy, hedge 25

1.12 Utility function 26

1.13 General objectives of game theory; Pascal's wager 27

2 Auctions and networks 29

2.1 Several players; the volunteers' dilemma 29

2.2 An example on iterated elimination of dominated strategies 31

2.3 Second price and increasing bid auctions 32

2.4 Escalating conflicts 34

2.5 Braess paradox 35

2.6 Wardrop equilibria and selfish routing 37

3 Wise men and businessmen 39

3.1 Wise men and their wives; imp in the bottle 39

3.2 King Solomon's wisdom 41

3.3 Chain store paradox; centipede game 43

3.4 Normal and extensive forms of a game; battle of the sexes revisited 46

3.5 Dynamic games and subgame perfection; pursuit games 48

3.6 Fair division and the ultimatum game 50

3.7 Cooperation by threat and punishment; infinitely repeated games 51

3.8 Computer tournaments; the triumph of the strategy Tit-for-Tat 58

3.9 Logical games; limits of the sequences 59

3.10 Russian Roulette; games with incomplete information 62

4 Hawk and doves, lions and lambs 65

4.1 Fitness and stability in population biology (general ideas) 65

4.2 Hawk and Dove games as social dilemmas 67

4.3 Mixed strategies, probability and chance 69

4.4 The theorems of Nash and von Neumann 73

4.5 Expectation and risk; St. Petersburg game 74

4.6 Symmetric mixed strategies Nash equilibria 75

4.7 Invasion of mutants and evolutionary stable strategies 77

4.8 The sex ratio game 80

5 Coalitions and distribution 81

5.1 Distribution of costs and gains; the core of the game 81

5.2 General principles of fair distribution 85

5.3 Utilitarianism and egalitarianism; compromise set 88

5.4 Equilibrium priced 91

5.5 Linear models and linear programming 93

6 Presidents and dictators 95

6.1 Collective choice; problems of voting 95

6.2 Four examples of voting rules 97

6.3 Criteria of quality of voting rules 99

6.4 The minority principle; dictators 102

7 At the doors of quantum games 105

7.1 Quantum bits and Schrödinger's cat 105

7.2 Lattices and quantum logic 108

7.3 Rendezvous of Bob and Alice 111

8 It's party time! 113

8.1 Combinatorial games 113

8.2 Addition and subtraction of games, order structure 117

8.3 Impartial games and Nim numbers 119

8.4 Games as numbers and numbers as games 123

Armed with mathematics 125

9 A rapid course in mathematical game theory 127

9.1 Three classical examples of Nash equilibria in economics 127

9.2 Mixed strategies for finite games 131

9.3 Evolutionary stable strategies 139

9.4 Replicator dynamics, Nash's fields and stability 141

9.5 Iterative method of solving matrix games 156

9.6 Zero-sum games and linear programming 159

9.7 Backward induction and dynamic programming 161

9.8 Cooperative games: Nucleus and the Shapley vector 167

9.9 Revision exercises 168

9.10 Solutions to revision exercises 173

10 Examples of game models 179

10.1 A static model of strategic investment 179

10.2 Variations Cournot's theme: Territorial price building 182

10.3 Models of inspection 184

10.4 A dynamic model of strategic investments 191

10.5 Game theoretic approach to the analysis of colored (or rainbow) options 197

11 Elements of more advanced analysis 211

11.1 Short overview 211

11.2 Two proofs of the Nash-Gliksberg theorem on the existence of equilibria 212

11.3 Introduction to structural stability 216

11.4 Introduction to abstract differential games 229

11.5 Cooperative games versus zero-sum games 237

11.6 Turnpikes for stochastic games 240

11.7 Games and tropical (or idempotent) mathematics 245

11.8 The first order partial differential equations in multi-criteria optimization problems 253

11.9 General flows of deterministic and stochastic replicator dynamics 259

Bibliography 273

Index 283