Understanding Game Theory : Introduction to the Analysis of Many Agent Systems with Competition and Cooperation

by Kolokoltsov, Vassili N.; Malafeyev, Oleg A.

• ISBN13: 9789814291712
• ISBN10: 9814291714
• Format: Hardcover
• Copyright: 2010-03-31
• Publisher: Textstream

Summary

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-renown 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.

Table of Contents

Preface	p. vii
Basic ideas	p. 1
Around the prisoner's dilemma	p. 3
What is a two-player game?	p. 3
Prisoner's dilemma. Dominated strategies and Pareto optimality	p. 7
Prisoner's dilemma for crooks, warriors and opera lovers	p. 9
Discrete duopoly models and common pool resources; public goods	p. 12
Common knowledge, rationality and iterated elimination of strictly dominated strategies	p. 13
Weak dominance; debtors and creditors	p. 15
Nash equilibrium	p. 16
Battle of the sexes and Kant's categorical imperative	p. 19
Chicken game and the Cuban missile crisis	p. 21
Social dilemmas	p. 22
Guaranteed payoff, minimax strategy, hedge	p. 25
Utility function	p. 26
General objectives of game theory; Pascal's wager	p. 27
Auctions and networks	p. 29
Several players; the volunteers' dilemma	p. 29
An example on iterated elimination of dominated strategies	p. 31
Second price and increasing bid auctions	p. 32
Escalating conflicts	p. 34
Braess paradox	p. 35
Wardrop equilibria and selfish routing	p. 37
Wise men and businessmen	p. 39
Wise men and their wives; imp in the bottle	p. 39
King Solomon's wisdom	p. 41
Chain store paradox; centipede game	p. 43
Normal and extensive forms of a game; battle of the sexes revisited	p. 46
Dynamic games and subgame perfection; pursuit games	p. 48
Fair division and the ultimatum game	p. 50
Cooperation by threat and punishment; infinitely repeated games	p. 51
Computer tournaments; the triumph of the strategy Tit-for-Tat	p. 58
Logical games; limits of the sequences	p. 59
Russian Roulette; games with incomplete information	p. 62
Hawk and doves, lions and lambs	p. 65
Fitness and stability in population biology (general ideas)	p. 65
Hawk and Dove games as social dilemmas	p. 67
Mixed strategies, probability and chance	p. 69
The theorems of Nash and von Neumann	p. 73
Expectation and risk; St. Petersburg game	p. 74
Symmetric mixed strategies Nash equilibria	p. 75
Invasion of mutants and evolutionary stable strategies	p. 77
The sex ratio game	p. 80
Coalitions and distribution	p. 81
Distribution of costs and gains; the core of the game	p. 81
General principles of fair distribution	p. 85
Utilitarianism and egalitarianism; compromise set	p. 88
Equilibrium priced	p. 91
Linear models and linear programming	p. 93
Presidents and dictators	p. 95
Collective choice; problems of voting	p. 95
Four examples of voting rules	p. 97
Criteria of quality of voting rules	p. 99
The minority principle; dictators	p. 102
At the doors of quantum games	p. 105
Quantum bits and Schrödinger's cat	p. 105
Lattices and quantum logic	p. 108
Rendezvous of Bob and Alice	p. 111
It's party time!	p. 113
Combinatorial games	p. 113
Addition and subtraction of games, order structure	p. 117
Impartial games and Nim numbers	p. 119
Games as numbers and numbers as games	p. 123
Armed with mathematics	p. 125
A rapid course in mathematical game theory	p. 127
Three classical examples of Nash equilibria in economics	p. 127
Mixed strategies for finite games	p. 131
Evolutionary stable strategies	p. 139
Replicator dynamics, Nash's fields and stability	p. 141
Iterative method of solving matrix games	p. 156
Zero-sum games and linear programming	p. 159
Backward induction and dynamic programming	p. 161
Cooperative games: Nucleus and the Shapley vector	p. 167
Revision exercises	p. 168
Solutions to revision exercises	p. 173
Examples of game models	p. 179
A static model of strategic investment	p. 179
Variations Cournot's theme: Territorial price building	p. 182
Models of inspection	p. 184
A dynamic model of strategic investments	p. 191
Game theoretic approach to the analysis of colored (or rainbow) options	p. 197
Elements of more advanced analysis	p. 211
Short overview	p. 211
Two proofs of the Nash-Gliksberg theorem on the existence of equilibria	p. 212
Introduction to structural stability	p. 216
Introduction to abstract differential games	p. 229
Cooperative games versus zero-sum games	p. 237
Turnpikes for stochastic games	p. 240
Games and tropical (or idempotent) mathematics	p. 245
The first order partial differential equations in multi-criteria optimization problems	p. 253
General flows of deterministic and stochastic replicator dynamics	p. 259
Bibliography	p. 273
Index	p. 283
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