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Combinatorial games : tic-tac-toe theory

Author: József Beck
Publisher: Cambridge : Cambridge University Press, 2008.
Series: Encyclopedia of mathematics and its applications, v. 114.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"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. This is the subject of combinatorial game theory. Most board games are a challenge for mathematics: to analyze a position one has to examine the  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: József Beck
ISBN: 9780521461009 0521461006
OCLC Number: 175284055
Description: xiv, 732 pages : illustrations ; 24 cm.
Contents: pt. A. Weak win and strong draw --
ch. I. Win vs. weak win --
Illustration : every finite point set in the plane is a weak winner --
Analyzing the proof of theorem 1.1 --
Examples : tic-tac-toe games --
More examples : tic-tac-toe like games --
Games on hypergraphs, and the combinatorial chaos --
ch. II. The main result : exact solutions for infinite classes of games --
Ramsey theory and clique games --
Arithmetic progressions --
Two-dimensional arithmetic progressions --
Explaining the exact solutions : a meta-conjecture --
Potentials and the Erdős-Selfridge theorem --
Local vs. global --
Ramsey theory and hypercube tic-tac-toe --
pt. B. Basic potential technique : game-theoretic first and second moments --
ch. III. Simple applications --
Easy building via theorem 1.2 --
Games beyond Ramsey theory --
A generalization of Kaplansky's game --
ch. IV. Games and randomness --
Discrepancy games and the variance --
Biased discrepancy games : when the extension from fair to biased works! --
A simple illustration of "randomness" (I) --
A simple illustration of "randomness" (II) --
Another illustration of "randomness" in games. pt. C. Advanced weak win : game-theoretic higher moment --
ch. V. Self-improving potentials --
Motivating the probabilistic approach --
Game-theoretic second moment : application to the picker-choose game --
Weak win in the lattice games --
Game-theoretic higher moments --
Exact solution of the clique game (I) --
More applications --
Who-scores-more games --
ch. VI. What is the biased meta-conjecture, and why is it so difficult? --
Discrepancy games (I) --
Discrepancy games (II) --
Biased games (I) : biased meta-conjecture --
Biased games (II) : sacrificing the probabilistic intuition to force negativity --
Biased games (III) : sporadic results --
Biased games (IV) : more sporadic results --
pt. D. Advanced strong draw : game-theoretic independence --
ch. VII. BigGame-SmallGame decomposition --
The Hales-Jewett conjecture --
Reinforcing the Erdős-Selfridge technique (I) --
Reinforcing the Erdős-Selfridge technique (II) --
Almost disjoint hypergraphs --
Exact solution of the clique game (II). ch. VIII. Advanced decomposition --
Proof of the second ugly theorem --
Breaking the "square-root barrier" (I) --
Breaking the "square-root barrier" (II) --
Van der Waerden game and the RELARIN technique --
ch. IX. Game-theoretic lattice-numbers --
Winning planes : exact solution --
Winning lattices : exact solution --
I-can-you-can't games --
second player's moral victory --
ch. X. Conclusion --
More exact solutions and more partial results --
Miscellany (I) --
Miscellany (II) --
Concluding remarks --
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.
Series Title: Encyclopedia of mathematics and its applications, v. 114.
Responsibility: József Beck.
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Abstract:

In this comprehensive volume, Jozsef Beck shows readers how to escape from the combinatorial chaos arising in the analysis of many games by using the fake probabilistic method, a game-theoretic  Read more...

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'... 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 Read more...

 
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