Given a set P of natural numbers, we consider infinite games where the winning condition is a regular ω-language parameterized by P. In this context, an ω-word, representing a play, has letters consisting of three components: The first is a bit indicating membership of the current position in P, and the other two components are the letters contributed by the two players. Extending recent work of Rabinovich we study here predicates P where the structure (N, +1, P) belongs to the pushdown hierarchy (or "Caucal hierarchy"). For such a predicate P where (N, +1, P) occurs in the k-th level of the hierarchy, we provide an effective determinacy result and show that winning strategies can be implemented by deterministic level-k pushdown automata.
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