Introduction to the theory of games. Concepts, methods, applications
The book is a revised and expanded version of the edition, published in 1985 (Budapest: Akademici Kiado, see the review in Zbl 0578.90096). It intends not only to give concepts and models of modern game theory but also to survey methods for seeking game solutions. Nonconvex problems are discuss as well. The book covers almost all classical results of game theory except of differential, hierarchial, and position games. Some special classes of games, such as stochastic and combinatorial are not covered. However, other special classes, such as duels and Harsanyi models, are considered. The graduate course text features of the book are enhanced by the addition of problems and exercises. The book consists of two parts. Part I is devoted to noncooperative games. It has 13 chapters. Chapter 1 is introductory and defines extensive and normal forms of a game and a game tree. Next three chapters are devoted to Nash equilibrium (definition, existence, and uniqueness). The authors do not bound themselves by convex case but investigate various generalizations of concavity, in particular, concavity of the sum of the pay-off functions. Classical mixed extensions of finite games are introduced in Chapter 5 where proper and perfect equilibria are also discussed. Computation of mixed equilibria is considered in Chapter 6. Well-known methods are briefly mentioned, but Scarf-Hansen algorithm is only covered in details. Special attention is given to oligopoly games (Chapter 7): multi- and single-product cases. For dynamic oligopolies Cournot procedure is described (without Cournot name). For linear statements, its convergence is proved to be equivalent to duopoly case. Two-person games are thoroughly considered in Chapters 8 - 11 in a classical way. In Chapters 12 and 13 non-classical branches of game theory are presented: repeated games and games with incomplete information in the sence of Harsanyi approach. Part II is devoted to cooperative games. It has 10 chapters (14 - 23). Chapter 14 is introductory and defines the notion of characteristic function. The core is defined and investigated in Chapter 15. The core axiomatization is given (without proof). Chapter 16 is devoted to von Neumann-Morgenstern solution. Next type of a game solution, the nucleous, is considered in Chapter 17. Its properties and methods of computation are disscussed. Also the axiomatization is given (without proof). The Shapley value is studied and axiomatized in Chapter 18. In Chapter 19, the authors briefly consider solution concepts taking into account bargaining processes while coalitions are formed. These are the kernel and the bargaining set. Chapter 20 seems to be an addition to Chapter 18. It applies the Shapley value to the problem of cost allocation. The cost game and the savings game are described. Their solutions are discussed in the connection with various allocation rools. Chapter 21 gives a brief description of a market game as an example of games without transferable utility. A sufficient condition for the core of a market game to be nonempty is proved. Chapters 22 and 23 coverts bargaining problems. In Chapter 22, the Nash bargaining solution is compared with the Kalai-Smorodinsky solution. Also, in this chapter, multicriterial decision making is treated from game theoretic point of view. Chapter 23 is limited to two-person bargaining processes. A multiperiod negotiations model is studied. Though the book is intended in the first instance for Ph.D. and graduate students, some chapters can also be of interest to researches.
scientific article; zbMATH DE number 1349349