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The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy

J. D. Williams
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

I adored this book when I was in high school back in the 1960s (it was one of the few dozen math books in our tiny high school library), and I think there are still plenty of high-school students and laymen who would like it today. It is aimed at non-scientists, and by present-day mathematical standards it is not a math book at all, but rather an algorithms book. It doesn’t use any mathematics beyond arithmetic, and doesn’t explain any of the algorithms, although it does give some plausibility arguments and heuristics to partly justify them. The present volume is a 1986 Dover unaltered reprint of the 1966 revised edition from McGraw-Hill.

The book covers only zero-sum two-person games, but is quite complete for those. The coverage is divided up according to the number of available strategies; thus it starts with \(2 \times 2\) games (two choices for each player), then goes through \(2 \times n\), \(3 \times 3\), and so on, up through \(4 \times n\). In each case the method of solution is laid out in steps and illustrated with numerous examples. Each example in the narrative has a funny story and picture that goes with it. For example, Example 12, “The Sports Kit” has a shaggy-dog story about playing Russian Roulette. The stories are the best part of the book. Each chapter also has a set of exercises which are the bare payoff matrices (with no story), with answers in the back of the book. There is a Miscellany chapter that touches on various topics that are not covered in the main narrative.

The original edition was in 1954. This 1966 revised edition differs by some corrections and the addition of a chapter on the “pivot method” for solving games, which is in fact the simplex method of linear programming presented as an algorithm for solving general \(m \times n\) games.

This book, despite its many fine qualities, is still very limited, and most introductory books today cover a much wider variety of games and have more mathematical content. This is not a text for a college-level game theory course, but would good for enrichment in high school or a non-majors course. A somewhat similar but more modern and more mathematical book is Stahl’s A Gentle Introduction to Game Theory (AMS, 1999). It lacks the pictures and funny stories, but also deals primarily in zero-sum two-person games, and is aimed low, using nothing beyond coordinate geometry.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Preface to the Revised Edition

Preface to the First Edition


Two-Strategy Games

Three-Strategy Games

Four-Strategy Games and Larger Ones


General Method of Solving Games