Combinatorial Games: Tic-Tac-Toe Theory

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Cambridge University Press, Mar 20, 2008 - Mathematics - 732 pages
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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.
  

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Contents

PART A WEAK WIN AND STRONG DRAW
15
Analyzing the proof of Theorem1 1
41
TicTacToe like games
59
Games on hypergraphs and the combinatorial chaos
72
exact solutions for infinite classes
91
Potentials and the ErdosSelfridge Theorem
146
Local vs Global
163
Ramsey Theory and Hypercube TicTacToe
172
Sacrificing the probabilistic intuition to force
418
Sporadic results
430
More sporadic results
439
ADVANCED STRONG DRAW GAMETHEORETIC
459
Reinforcing the ErdosSelfridge technique I
470
Reinforcing the ErdosSelfridge technique II
479
Almost Disjoint hypergraphs
485
Exact solution of the Clique Game II
492

PART B BASIC POTENTIAL TECHNIQUE GAME
193
Games beyond Ramsey Theory
204
A generalization of Kaplanskys game
216
Games and randomness
230
when the extension from fair
245
A simple illustration of randomness I
260
A simple illustration of randomness II
270
Another illustration of randomness in games
286
ADVANCED WEAK WIN GAMETHEORETIC
305
application to
320
Weak Win in the Lattice Games
329
Gametheoretic higher moments
340
Exact solution of the Clique Game I
352
More applications
362
Whoscoresmore games
372
What is the Biased MetaConjecture and why is
380
Discrepancy games II
392
Biased MetaConjecture
400
Advanced decomposition
504
Breaking the squareroot barrier I
525
Breaking the squareroot barrier II
536
Van der Waerden Game and the RELARIN technique
545
Gametheoretic latticenumbers
552
exact solution
575
Conclusion
610
Miscellany I
620
Miscellany II
634
Concluding remarks
644
Appendix A Ramsey Numbers
658
Shelahs proof
669
A formal treatment of Positional Games
677
An informal introduction to game theory
705
Complete list of the Open Problems
716
What kinds of games? A dictionary
724
References
730
Copyright

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About the author (2008)

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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