Book contents
- Frontmatter
- Contents
- Preface
- A summary of the book in a nutshell
- PART A WEAK WIN AND STRONG DRAW
- PART B BASIC POTENTIAL TECHNIQUE – GAME-THEORETIC FIRST AND SECOND MOMENTS
- PART C ADVANCED WEAK WIN – GAME-THEORETIC HIGHER MOMENT
- PART D ADVANCED STRONG DRAW – GAME-THEORETIC INDEPENDENCE
- 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
- Complete list of the Open Problems
- What kinds of games? A dictionary
- Dictionary of the phrases and concepts
- References
PART B - BASIC POTENTIAL TECHNIQUE – GAME-THEORETIC FIRST AND SECOND MOMENTS
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- A summary of the book in a nutshell
- PART A WEAK WIN AND STRONG DRAW
- PART B BASIC POTENTIAL TECHNIQUE – GAME-THEORETIC FIRST AND SECOND MOMENTS
- PART C ADVANCED WEAK WIN – GAME-THEORETIC HIGHER MOMENT
- PART D ADVANCED STRONG DRAW – GAME-THEORETIC INDEPENDENCE
- 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
- Complete list of the Open Problems
- What kinds of games? A dictionary
- Dictionary of the phrases and concepts
- References
Summary
Part B is a practice session for the potential technique, demonstrating the enormous flexibility of this technique.
We look at about a dozen amusing “little” games (similar to the S-building game in Section 1). There is a large variety of results, starting with straightforward applications of Theorem 1.2 (“building”) and Theorem 1.4 (“blocking”), and ending with sophisticated proofs like the 6-page-long proof of Theorem 20.3 (“Hamiltonian cycle game”) and the 10-page-long proof of Theorem 15.1 (“Kaplansky's Game”).
The core idea is the mysterious connection between games and randomness. By using the terms “game-theoretic first moment” and “game-theoretic second moment,” we tried to emphasize this connection.
The point is to collect a lot of “easy” proofs. To get a “feel” for the subject the reader is advised to go through a lot of easy stuff. Reading Part B is an ideal warmup for the much harder Parts C-D.
A reader in a big rush focusing on the exact solutions may skip Part B entirely, and jump ahead to Sections 23–24 (where the “hard stuff” begins).
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- Combinatorial GamesTic-Tac-Toe Theory, pp. 193 - 194Publisher: Cambridge University PressPrint publication year: 2008