Combinatorial GamesTic-Tac-Toe TheorySeries: Encyclopedia of Mathematics and its Applications (No. 114)József BeckRutgers University, New Jersey


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, 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 exact results about infinite classes of many games, leading to the discovery of some striking new duality principles.

• 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
ContentsPreface; A summary of the book in a nutshell; Part A. Weak Win and Strong Draw: 1. Win vs. weak win; 2. The main result: exact solutions for infinite classes of games; Part B. Basic Potential Technique - Game-Theoretic First and Second Moments: 3. Simple applications; 4. Games and randomness; Part C. 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 D. Advanced Strong Draw - Game-theoretic Independence: 7. BigGame-SmallGame decomposition; 8. Advanced decomposition; 9. Game-theoretic lattice-numbers; 10. Conclusion; Complete list of open problems; What kind of games? A dictionary; Dictionary of the phrases and concepts; 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; References