Combinatorial Games

Tic-Tac-Toe Theory
Erscheinungsjahr 2009
ISBN: 978-0-521-46100-9
Verlag: Cambridge University Press

Buch, Englisch, Band 114, 746 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 1274 g

Reihe: Encyclopedia of Mathematics and its Applications

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.

Autoren/Hrsg.

Beck, Jozsef

Beck, József

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik Mathematik Operations Research Spieltheorie

Inhaltsverzeichnis

Preface; 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.

Über Autor(innen)

Beck, József
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.