Mathematics has been used (to varying degrees of success) to analyze games, usually with the intent of finding a “winning strategy.” This book gives a summary of some of the results. The author divides the universe of games into three categories: games of chance (games, such as roulette, whose uncertainty comes from random influences); combinatorial games (games, such as chess and go, whose uncertainty relies on the multiplicity of possible moves); and strategic games (games, such as poker and rock-paper-scissors, whose uncertainty lies in a lack of complete information). The book is divided into three sections, based on these categories.
The section on Games of Chance deals with several games, and uses these to introduce a great deal of probability theory: the basic theory, expectation and variance, the normal distribution, the Poisson distribution, the Monte Carlo method, Markov chains. As expected, it is difficult to derive winning strategies for games of chance — although there is a section which presents a method for “counting cards” in blackjack.
The second section, Combinatorial Games, delves into the notion of game theory, with Zermelo’s Theorem for two-person zero-sum games with perfect information. Several games and their winning strategies are studied, for example, NIM, go, and some of the classic games created by John Conway. This section culminates into a discussion of topics such as artificial intelligence, Turing machines, Gödel’s Incompleteness Theorem, and P-NP-PSPACE-EXPTIME problems. The final section deals with Strategic Games. Here the author considers whether psychology can be more effective than random chance (for example, bluffing in poker). Applying the notion of the minimax value to this picture results in a linear programming problem, and the introduction of the simplex method.
Given the wide variety of (fairly deep) mathematics mentioned here, you can’t expect the book to go into too much detail; even coming in at just under 500 pages, there isn’t enough room to cover these kinds of topics with any depth. However, the book does provide a remarkable summary, replete with numerous footnotes. (Since this book is the English version of a German text, roughly half of the references are German magazines, books, or journals.)
Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at email@example.com.