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Combinatorial Games
Tic-Tac-Toe Theory


Part of Encyclopedia of Mathematics and its Applications

  • Date Published: April 2011
  • availability: In stock
  • format: Paperback
  • isbn: 9780521184755

£ 69.99

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About the Authors
  • 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.

    • Learn how to escape the combinatorial chaos using the fake probabilistic method to analyze games such as tic-tac-toe, hex and solitaire
    • Unique and comprehensive text by the master of combinatorial game theory: describes striking results and new duality principles
    • With nearly 200 figures, plus many exercises and worked examples, the concepts are made fully accessible
    • Now updated to reflect new results in the field since its original publication in 2008
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    Reviews & endorsements

    '… this book is a milestone in Game Theory, it will become a classic …' Acta Scientiarum Mathematicarum

    '… a most thorough and useful treatment of the subject (so far insufficiently presented in the literature) with an enormous store of results, links with other theories, and interesting open problems.' A. Pultr, Mathematical Reviews

    '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

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    Product details

    • Date Published: April 2011
    • format: Paperback
    • isbn: 9780521184755
    • length: 750 pages
    • dimensions: 234 x 156 x 37 mm
    • weight: 1.09kg
    • contains: 170 b/w illus. 40 exercises
    • availability: In stock
  • Table of Contents

    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

  • Author

    József Beck, Rutgers University, New Jersey
    József Beck is a Professor in the Mathematics Department of Rutgers University. He has received the Fulkerson Prize for Research in Discrete Mathematics and has written around 100 research publications. He is the co-author, with W. L. Chen, of the pioneering monograph Irregularities of Distribution.

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