  We study two-player zero-sum games over infinite-state graphs with
boundedness conditions. Our first contribution is about the strategy
complexity, i.e the memory required for winning strategies: we prove that over
general infinite-state graphs, memoryless strategies are sufficient for
finitary B\"uchi games, and finite-memory suffices for finitary parity games.
We then study pushdown boundedness games, with two contributions. First we
prove a collapse result for pushdown omega B games, implying the decidability
of solving these games. Second we consider pushdown games with finitary parity
along with stack boundedness conditions, and show that solving these games is
EXPTIME-complete.
