The objective of game-playing is winning, but very often winning is impossible for the simple reason that the game is a draw game: either player can force a draw. Blocking the opponent's winning sets is a solid way to force a draw; this is what we call a Strong Draw.

The main issue here is the Neighborhood Conjecture. The general case remains unsolved, but we can prove several useful partial results about blocking (called the Three Ugly Theorems).

Our treatment of the blocking part has a definite architecture. Metaphorically speaking, it is like a five-storied building where Theorems 34.1, 37.5, 40.1 represent the first three floors in this order, and Sections 43 and 44 represent the fourth and fifth floors; the higher floors are supported by the lower floors (there is no shortcut!).

An alternative way to look at the Neighborhood Conjecture is the Phantom Decomposition Hypothesis (see the end of Section 19), which is a kind of gametheoretic independence. In fact, there are two interpretations of game-theoretic independence: a “trivial” interpretation and a “non-trivial” one.

The “trivial” (but still very useful) interpretation is about disjoint games; Pairing Strategy is based on this simple observation. Disjointness guarantees that in each component either player can play independently from the rest of the components.

In the “non-trivial” interpretation the initial game does not fall apart into disjoint components.