Introducing Game Theory and Its Applications

The author’s aim was to present an easy-to-read introduction to the basic ideas and techniques of game theory and the possibilities of its applications. The book is divided into 4 chapters. Chapter 1 introduces the reader to combinatorial games. A combinatorial game is defined from the point of view of the traditional classification as a finite two-person zero-sum game with perfect information and deterministic moves. The fundamental theorem for combinatorial games by Zermelo is proved, and examples of some simpler combinatorial games and paying techniques are presented. Chapters 2 and 3 contain the traditional theory of two-person zero-sum games and usual solution methods. Some of them are designed for games with a special structure (e.g. (m x n)-matrix games, in which either m or n is equal to 2 or m=n=2). After having introduced the concept of linear programming and presented the main ideas of the simplex method and the duality theory, the author explains how a general (m x n)-matrix game can be solved by the simplex method applied to a pair of special dual linear programming problems. Chapter 4 includes some fundamental approaches to solving non-zero-sum cooperative games with more than two players. Such concepts as the Nash equilibrium, Nash arbitration procedures, the Shapley value, imputations, and stable sets are defined and their main properties are investigated. The text includes numerical exercises as well as examples of applications. Since the understanding of some proofs and methods requires the knowledge of the finite probability theory, some basic facts from this theory are included in the first appendix attached to the text of the book after Chapter 4. The other two appendices contain some results from the utility theory and Nash’s theorem. The book is concluded with answers to selected exercises and an extensive bibliography. It can be recommended to readers with a limited mathematical knowledge who are interested in game theory and its applications in economics, political science and biology.

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