ADVANCED WEAK WIN – GAME-THEORETIC HIGHER MOMENT
József Beck · Cambridge University Press eBooks · 2008
Here is a nutshell summary of what we did in Part A: the goal of the first chapter was to introduce the basic concepts such as Positional Game, Weak Win, Strong Draw, and to demonstrate the power of the potential technique on several amusing examples. The goal of the second chapter was to formulate the main results such as Theorem 6.4 and Theorem 8.2 (“exact solutions”), and also the Meta-Conjecture, the main issue of the book. Part B was a practice session for the potential technique. In the forthcoming Parts C–D, we discuss the most difficult proofs, in particular the exact solutions of our Ramseyish games with 2-dimensional goals. Part C is the building part and Part D is (mainly) the blocking part. In Part A, we introduced two simple “linear” criterions (Theorem 1.2 and Theorem 1.4), and gave a large number of applications. Here, in Part C, we develop some more sophisticated “higher moment” criterions. The motivation for “higher moment” comes from Probability Theory. The “higher moment” criterions are applied in a way very similar to how some of the main results of classical Probability Theory – such as the central limit theorem and the law of the iterated logarithm – are all based on higher moment techniques. Note in advance that the last part of the book (Part D) also has a strong probabilistic flavor: Part D is about how to “sequentialize” the global concept of statistical independence .