. L O ] 1 9 M ay 1 99 9 FILTERS AND GAMES

We obtain game–theoretic characterizations for meagerness and rareness of filters on ω. One of the classical methods for obtaining a set of real numbers which does not have the property of Baire, is to interpret appropriate filters on N = {1,2,3, . . . } as subsets of [0, 1]. Filters which result in a set having the property of Baire have nice combinatorial characterizations, due to Talagrand [3]: Theorem 1 (Talagrand). For a filter F on ω the following are equivalent: 1. F does not have the property of Baire, 2. The set of enumeration functions of sets in F is unbounded in ω, ordered by eventual domination, 3. For every partition of ω into disjoint finite sets, there is a set in F which is disjoint from infinitely many blocks in the partition. In particular, we see that a filter is meager if, and only if, it has the property of Baire. We use this without further notice below. These combinatorial characterizations suggest certain infinite two–player games. We study such a game in section 1. We prove that having the property of Baire is equivalent with the assertion that player ONE of our game has a winning strategy (Theorem 3). In the second section we study a natural variation of the first game, and show that here, being non-rare is equivalent to ONE having a winning strategy in this game (Theorem 5). Our notation is mostly standard; the only exception may be that we use ⌢ to denote concatenation of sequences. From now on, let F ⊂ P(N) be a non–principal filter.