Introducing Game Theory and its Applications

Rating:
Summary: approachable Introduction
Review: Introducing Game Theory and its Applications by Elliot Mendelson (Chapman & Hall/CRC) The mathematical study of games is an intriguing endeavor with implications and applications that reach far beyond tic-tac-toe, chess, and poker to economics, business, and even biology. Most texts on the subject, however, are written at the graduate level for those with strong mathematics, economics, or business backgrounds.

In a clear and refreshing departure from this trend, Introducing Game Theory and Its Applications presents an easy-to-read introduction to the basic ideas and techniques of game theory. After a brief introduction, the author begins with a chapter devoted to combinatorial games-a topic neglected or treated minimally in most other texts. The focus then shifts to two-person zero-sum games and their solution. Here the author presents the simplex method, based on linear programming, for solving these games and develops within his presentation the required background in linear programming. The final chapter presents some of the fundamental ideas and tools of non-zero-sum games and games with more than two players, including an introduction to cooperative game theory.

Excerpt: This book will not only satisfy the curiosity of those whose interest in the subject was piqued by the 1994 Nobel Prize awarded to Harsanyi, Nash, and Selten. It also prepares its readers for more advanced study of game theory's applications in economics, business, and the physical, biological, and social sciences.
This book is intended to be an easy-to-read introduction to the basic ideas and techniques of game theory. It can be used as a class textbook or for self-study. It also could be helpful for students who must learn some game theory in a course in a related subject (such as microeconomics) and have limited mathematical background.
In recent years, and especially since the Nobel Prize in Economics was awarded in 1994 to John C. Harsanyi, John F. Nash Jr., and Reinhard Selten for their research in game theory, people have been intrigued by the unusual connection between games and mathematics, and the author hopes that this curiosity may be satisfied to some extent by reading this book. The book also will prepare them for deeper study of applications of game theory in economics and business and in the physical, biological, and social sciences.
The first part of the text (Chapter 1) is devoted to combinatorial games. These games tend to be more recreational in nature and include board games like chess and checkers (and even a simple children's game like Tic-Tac-Toe). There are also many such games that are challenging even for accomplished mathematicians and our study covers various clever techniques for successful play. The rest of the book deals with the general theory of games. Chapters 2 and 3 contain a thorough treatment of two-person zero-sum games and their solution, which is the most well-understood part of game theory and was developed from the 1920s through the 1950s. John von Neumann was responsible, almost by himself, for inventing the subject, which reached its climax with the elaboration of the simplex method. Chapter 4 introduces the reader to games that are not zero-sum and/or involve more than two players. Here it is often natural to consider cases where the players must no longer act as isolated individuals but are permitted to form coalitions with other players. Games in which this happens are called cooperative games. All of this is an area of current research and there is still no general consensus about its concepts, methods, and applications. Here we have only attempted to present the basic ideas and some of the less controversial results, so that readers can venture further on their own. The reader will find in this chapter some of the applications that make game theory so interesting, for example, in economics, in the theory of political power, and in evolutionary biology. We assume no previous acquaintance with the details of these subjects. Our treatment is intended to provide enough of the fundamental concepts and
techniques of game theory to make it easier to understand more advanced applications.
There are three Appendices. Appendix 1 reviews finite probability theory and provides whatever is necessary to understand all applications of probability in the book. Appendix 2 sketches an axiomatic treatment of utility theory. Although the utility concept is necessary for a proper understanding of game theory (and economics in general), we have avoided dealing with the subject in the body of the text because it would be a difficult and unnecessary distraction for the readers in their first confrontation with game theory. Appendix 3 contains two proofs of Nash's Theorem on the existence of Nash equilibria. The reason for consigning them to an appendix is that they depend on sophisticated results from topology, with which many readers will not be familiar.
The section of Answers to Selected Exercises contains brief solutions to enough of the exercises for the readers to be sure that they understand what is going on, and then usually leaves some exercises to be answered on their own. There is an extensive Bibliography that contains not only the articles and books referred to in the text, but also many readings that may attract the readers' interest and extend their knowledge.