The following technical report is available from http://aib.informatik.rwth-aachen.de:
Parametrized Regular Infinite Games and Higher-Order Pushdown Strategies Paul Hänsch, Michaela Slaats, Wolfgang Thomas AIB 2009-18
Given a set $P$ of natural numbers, we consider infinite games where the winning condition is a regular $\omega$-language parametrized by $P$. In this context, an $\omega$-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 $(\mathbb{N}, +1, P)$ belongs to the pushdown hierarchy (or ``Caucal hierarchy''). For such a predicate $P$ where $(\mathbb{N}, +1, P)$ 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.