Appendix C - A formal treatment of Positional Games

Summary

Everything that we know about ordinary win in a positional game comes from Strategy Stealing. We owe the reader a truly precise treatment of this remarkable existence argument. Also we make the vague term “exhaustive search” precise by introducing a backtracking algorithm called “backward labeling”. We start the formal treatment with a definite terminology (which is common sense anyway).

Terminology of Positional Games. There are some fundamental notions of games which are used in a rather confusing way in everyday language. First, we must distinguish between the abstract concept of a game, and the individual plays of that game.

In everyday usage, game and play are often synonyms. Tennis is a good example for another kind of confusion. To play a game of tennis, we have to win two or three sets, and to win a set, we must win six (or seven) games; i.e., certain components of the game are again called “games.” If the score in a set is 6:6 – a “tie” – then, by a relatively new rule in tennis, the players have to play a “tie-break.” We will avoid “tie,” and use “draw” instead; “drawing strategy” sounds better than “tie, or tying, strategy.”

In our terminology a game is simply the set of the rules that describe it.