Combinatorial Games
Tic-Tac-Toe Theory

A comprehensive and unique volume by the master of combinatorial game theory.

József Beck (Author)

9780521461009, Cambridge University Press

Hardback, published 20 March 2008

748 pages, 170 b/w illus. 40 exercises
23.7 x 16.4 x 4.4 cm, 1.21 kg

'This seems to be the best and most useful treatment of the subject so far … The book is recommended for a broad mathematical audience. Almost all concepts from other parts of mathematics are explained so it is convenient both for the specialist seeking a detailed survey of the topic and for students hoping to learn something new about the subject. The book has a potential to become a milestone in the development of combinatorial game theory.' EMS Newsletter

Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication.

Preface
A summary of the book in a nutshell
Part I. Weak Win and Strong Draw: 1. Win vs. weak win
2. The main result: exact solutions for infinite classes of games
Part II. Basic Potential Technique – Game-Theoretic First and Second Moments: 3. Simple applications
4. Games and randomness
Part III. Advanced Weak Win – Game-Theoretic Higher Moment: 5. Self-improving potentials
6. What is the Biased Meta-Conjecture, and why is it so difficult?
Part IV. Advanced Strong Draw – Game-Theoretic Independence: 7. BigGame-SmallGame decomposition
8. Advanced decomposition
9. Game-theoretic lattice-numbers
10. Conclusion
Appendix A. Ramsey numbers
Appendix B. Hales–Jewett theorem: Shelah's proof
Appendix C. A formal treatment of positional games
Appendix D. An informal introduction to game theory
Appendix E. New results
Complete list of the open problems
What kinds of games? A dictionary
Dictionary of the phrases and concepts
References.

Subject Areas: Combinatorics & graph theory [PBV], Mathematical foundations [PBC]