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
- Chapter V Self-improving potentials
- Chapter VI What is the Biased Meta-Conjecture, and why is it so difficult?
- 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
Chapter V - Self-improving potentials
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
- Chapter V Self-improving potentials
- Chapter VI What is the Biased Meta-Conjecture, and why is it so difficult?
- 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
In Chapter IV, we start to explore the connection between randomness and games. A more systematic study is made of the probabilistic approach, that is actually refered to as a “fake probabilistic method.”
The main ingredients of the “fake probabilistic method” are:
(1) the two linear criterions (“Part A”) – for some applications see Part B;
(2) the advanced Weak Win criterion together with the ad hoc method of Section 23 (“Part C”);
(3) the BigGame–SmallGame Decomposition and its variants (“Part D”).
The main result in Chapter V is (2): the Advanced Weak Win Criterion, a complicated “higher moment” criterion. It is complicated in many different ways:
(i) the form of the criterion is already rather complicated;
(ii) the proof of the criterion is long and complicated;
(iii) the application to the Clique Game requires complicated calculations.
This criterion basically solves the building part of the Meta-Conjecture (see Section 9).
Motivating the probabilistic approach
Let us return to Section 6: consider the Maker-Breaker version of the (KN, Kq) Clique Game (we don't use the notation [KN, Kq] any more). How do we prove lower bound (6.1)? How can Maker build such a large clique?
Halving Argument. The Ramsey criterion Theorem 6.2, combined with the Erdős–Szekeres bound, gives the size q = ½ log2N, which is roughly ¼ of the truth.
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- Combinatorial GamesTic-Tac-Toe Theory, pp. 307 - 379Publisher: Cambridge University PressPrint publication year: 2008