This book was first published in 1959, and as such does not reflect the large number of significant results in game theory that have taken place since then. At the time of publication, the theory of Nash equilibria was known but had not yet taken hold, the role of artificial intelligence in the theory of games was yet to unfold, and computer simulations were just beginning to be used to understand optimal strategies in games. These concepts and the computer now play a major role in game theory, and a modern book would reflect these developments more than this one. That being said, one still might peruse this book to get an idea of the history of game theory and possibly to appreciate the conceptual situation that existed at the time of publication. Frequently older books in mathematics stress more of the underlying intuition behind a subject than modern ones. Thus one could justify a reading of this book with this in mind, and possibly use it also as a reference and as a source of problem sets for those individuals teaching game theory in the classroom.
The book treats what is now called "classical" game theory, which is taken to be the time before the contributions of John Nash circa 1950-1953. The research in game theory in the classical period was dominated by the results of J. Von Neumann, O. Morgenstern, H.Kuhn, A. Tucker, and many researchers in the Rand Corporation. The mathematical techniques used were primarily drawn from linear algebra, analysis, and linear and nonlinear programming. Economic modeling, military strategies, and management problems provided the stimulus to this research. Indeed, mathematical economics has one of its roots in game theory. The very early work in game theory by James Waldegrave in the 1700s and Augustin Cournot in the 1800s is not discussed by the author. The work by Cournot on duopolies could be considered to be a version of Nash equilibrium.
The book is divided into two parts, with the first one treating matrix games, linear and nonlinear programming and mathematical economics, and the second the theory of infinite games. My interest in the book stemmed from being asked to look at a mathematical formulation of poker. The game of poker is considered by the author in a few places in the book, both in the context of discrete games and in infinite games. The case of infinite games is particularly interesting in that it involves the use of integral equations and fixed-point theorems.
Some of the more esoteric techniques from mathematics used in the book include a few from algebraic topology: the Brouwer and Kakutani fixed-point theorems; from real analysis: the Haar measure, which comes into play when considering the invariance of optimal strategies under the action of a compact transformation group (the idea of weak* convergence also is used here), and the Fourier transform, which appears in the consideration of bell-shaped kernels for infinite games.
Some of the major results in classical game theory that are proved in the book are: the Kuhn-Tucker theorem of nonlinear (concave) programming (and an example dealing with portfolio selection), the duality theorem for both linear and nonlinear programming (a connection with the law of supply and demand is discussed), the von Neumann model of an expanding economy (this gives an excellent introduction to dynamical modeling in economics),and a fairly lengthy overview of equilibrium in economics and its stability. In modern treatments this is best done using the techniques of differential topology. The author's treatment can be viewed as an elementary introduction to the more modern treatments.
