The tactical variation of the fundamental theorem

We prove the tactical variation of the fundamental theorem of finite games—for finite games with sufficiently rich board positions, one of the players has a winning tactic or both have drawing tactics

This is part two of a series on tactics and strategies in the theory of games.

In last week’s post, we introduced the distinction between tactics and strategies in the theory of games, observing that many familiar games, including Nim, Connect Four, tic-tac-toe, and Othello, admit the conclusion of the fundamental theorem of finite games for tactics as well as strategies—either one of the players has a winning tactic or both have drawing tactics.

But we also found games for which this conclusion was not the case, such as the Chocolatier’s game, as well as a simple finite game and (in the exercises) a finite variation of the Chocolatier’s game, with a bitter, forbidden chocolate.

This week, I should like to address the question of which games admit a positive tactical variation of the fundamental theorem. Can we find a general criterion that will suffice to know that one player has a winning tactic or both have drawing tactics?

Yes, indeed we can. Despite the negative examples of the previous essay, I shall nevertheless prove a successful tactical version of the fundamental theorem of finite games, for finite games in which the board positions are somewhat fuller with information.