  Stochastic games combine controllable and adversarial non-determinism with
stochastic behavior and are a common tool in control, verification and
synthesis of reactive systems facing uncertainty. Multi-objective stochastic
games are natural in situations where several - possibly conflicting -
performance criteria like time and energy consumption are relevant. Such
conjunctive combinations are the most studied multi-objective setting in the
literature. In this paper, we consider the dual disjunctive problem. More
concretely, we study turn-based stochastic two-player games on graphs where the
winning condition is to guarantee at least one reachability or safety objective
from a given set of alternatives. We present a fine-grained overview of
strategy and computational complexity of such \emph{disjunctive queries} (DQs)
and provide new lower and upper bounds for several variants of the problem,
significantly extending previous works. We also propose a novel value
iteration-style algorithm for approximating the set of Pareto optimal
thresholds for a given DQ.
