Meager-nowhere dense games (I): N-tactics

In the introduction to this article we give a brief survey of a problem in the theory of Banach-Mazur games. We introduce two games, MG(J) and SMG(J) (where J is a free ideal on some set), which evolved from a study of an example relevant to this problem. The second player has a winning perfect information strategy in both of these games and we examine under what conditions it suffices for the second player to remember only the most recent n or fewer moves of the opponent (n some fixed positive integer) in order to insure a win. Strategies depending on only this information are called n-tactics.