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 VI - What is the Biased Meta-Conjecture, and why is it so difficult?
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
There are two natural ways to generalize the concept of Positional Game: one way is the (1) discrepancy version, where Maker wants (say) 90% of some hyperedge instead of 100%. Another way is the (2) biased version like the (1 : 2) play, where underdog Maker claims 1 point per move and Breaker claims 2 points per move.
Chapter VI is devoted to the discussion of these generalizations.
Neither generalization is a perfect success, but there is a big difference. The discrepancy version generalizes rather smoothly; the biased version, on the other hand, leads to some unexpected tormenting(!) technical difficulties.
The main issue here is to formulate and prove the Biased Meta-Conjecture. The biased case is work in progress; what we currently know is a bunch of (very interesting!) sporadic results, but the general case remains wide open.
We don't see any a priori reason why the biased case should be more difficult than the fair (1:1) case. No one understands why the general biased case is still unsolved.
The Biased Meta-Conjecture is the most exciting research project that the book can offer. We challenge the reader to participate in the final solution.
The biased Maker–Breaker and Avoider–Forcer games remain mostly unsolved, but we are surprisingly successful with the biased (1:s) Chooser–Picker game where Chooser is the underdog (in each turn Picker picks (s + 1) new points, Chooser chooses one of them, and the rest goes back to Picker).
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- Information
- Combinatorial GamesTic-Tac-Toe Theory, pp. 380 - 458Publisher: Cambridge University PressPrint publication year: 2008