  We consider a class of infinite-state stochastic games generated by stateless
pushdown automata (or, equivalently, 1-exit recursive state machines), where
the winning objective is specified by a regular set of target configurations
and a qualitative probability constraint `>0' or `=1'. The goal of one player
is to maximize the probability of reaching the target set so that the
constraint is satisfied, while the other player aims at the opposite. We show
that the winner in such games can be determined in PTIME for the `>0'
constraint, and both in NP and coNP for the `=1' constraint. Further, we prove
that the winning regions for both players are regular, and we design algorithms
which compute the associated finite-state automata. Finally, we show that
winning strategies can be synthesized effectively.
