Chance, logic and bluff. Mathematics in games: methods, results and limits. (Glück, Logik und Bluff. Mathematik im Spiel: Methoden, Ergebnisse und Grenzen.)

[B] Wiesbaden: Vieweg. xiv, 357 S. DM 49.80 (1998). ISBN 3-528-06997-X/pbk

The book under review is a fairly unusual one. One way of describing it is the following. It provides an introduction to the basic concepts in the theory of probability, up to and including Markov chains, by means of a large variety of games containing a considerable component of randomness, such as rolling dice, roulette and so forth. It further provides an introduction to elements of the mathematical theory of games and to complexity theory by means of (mainly) combinatorial games such as chess, go, Nim, backgammon, and Mastermind, and goes on to more advanced aspects of game theory and optimisation (including linear programming) with poker as the paradigmatic strategic game. An alternative description could be given by saying that the book deals with a wealth of different (type of) games by uncovering the role of various mathematical structures that are inherent in these games and are necessary or helpful for a deeper understanding of them.

Irrespective of which perspective one chooses to adopt, the book is rich, informative and stimulating. It pays much attention to explanation of both the mathematics and the games dealt with. Every now and then such expanations are given in separate boxes. There are multitudes of historical comments, and an extensive collection of references to primary research papers and monographs, to expository books, including textbooks, and to recreational and popularisation publications. Any opportunity to point to and speak about mathematics hidden in or invoked by the material covered is utilised, sometimes even to the degree of digression.

Although the book is in a way non-technical, in that it assumes no prior knowledge of probability theory, combinatorics, game theory, LP, etc. it does in fact presuppose a basic knowledge of mathematics, including terminology, symbolism and, above all, mathematical thinking. So the potential readership needs to know general mathematics at, say, a beginning tertiary level. To readers with these prerequisites -- and with the ability to read German and a moderate to strong interest in games -- the book is highly recommended.

M.Niss (Roskilde)