This is a new edition and translation into English of the German original published in 2001. The preface contains a brief discussion of the role of uncertainty in games, which leads to a division of games into three main types: games of chance, in which chance is more influential than decisions of the players; games whose uncertainty rests on the large number of possible moves, called combinatorial games; and strategic games, where uncertainty arises primarily from the fact that not all players have the same information about the current state of the game. Some games fall clearly into one of these categories (roulette, chess, rock-paper-scissors), but most combine features of two or all three. The book is divided into three parts, according to these game types, and each part into some 15 chapters, each devoted to a single problem, usually a game. The chapters are each introduced by an interesting question, for example: “It is hardly to be expected that in 37 spins of the roulette wheel, all 37 numbers will appear exactly once. How many different numbers will appear on average?” Among the games discussed, in addition to those already mentioned, are dice, lotteries, poker, backgammon, Risk, Monopoly, snakes and ladders, blackjack, chess, nim, go, and baccarat.
The aim is to introduce the mathematics that will allow analysis of the problem or game. This is done in gentle stages, from chapter to chapter, so as to reach as broad an audience as possible. The opening chapters introduce the basic concepts of elementary probability, building from there to random variables, Markov chains and some statistics by the end of Part I. In Parts II and III we find discussions of strategy, the minimax theorem, algorithms, the halting problem, Gödel’s Incompleteness Theorem, complexity theory, and linear optimization. Anyone who likes games and has a taste for analytical thinking will enjoy this book, and for those who wish to go deeper there are plenty of suggestions for further reading.
Peter Fillmore, Dalhousie University, Halifax, NS
CMS Notes, published by the Canadian Mathematical Society, 37 (4) (May 2005), p.9.