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 available options, and then the further options available after selecting any option, and so on. This leads to combinatorial chaos, where brute force study is impractical.
In this comprehensive volume, Jozsef 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.
A summary of the book in a nutshell 1
Pt. A Weak Win and Strong Draw 15
Ch. I Win vs. Weak Win 17
1 Illustration: every finite pointset in the plane is a Weak Winner 19
2 Analyzing the proof of Theorem 1.1 32
3 Examples: Tic-Tac-Toe games 42
4 More examples: Tic-Tac-Toe like games 59
5 Games on hypergraphs, and the combinatorial chaos 72
Ch. II The main result: exact solutions for infinite classes of games 91
6 Ramsey Theory and Clique Games 92
7 Arithmetic progressions 106
8 Two-dimensional arithmetic progressions 118
9 Explaining the exact solutions: a Meta-Conjecture 131
10 Potentials and the Erdos-Selfridge Theorem 146
11 Local vs. Global 163
12 Ramsey Theory and Hypercube Tic-Tac-Toe 172
Pt. B Basic Potential Technique - Game-Theoretic First and Second Moments 193
Ch. III Simple applications 195
13 Easy building via Theorem 1.2 196
14 Games beyond Ramsey Theory 204
15 A generalization of Kaplansky's game 216
Ch. IV Games and randomness 230
16 Discrepancy Games and the variance 231
17 Biased Discrepancy Games: when the extension from fair to biased works! 245
18 A simple illustration of "randomness" (I)
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