Covering games and the Banach-Mazur game: k-tactics

Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasing sequence of elements of the sigma completion of J, and TWO tries to cover the union of this sequence by playing one set at a time from J. We describe various conditions under which player TWO has has a winning strategy that uses only information about the most recent k moves of ONE, and apply some of these results to the Banach-Mazur game.