  Decisiveness has proven to be an elegant concept for denumerable Markov
chains: it is general enough to encompass several natural classes of
denumerable Markov chains, and is a sufficient condition for simple qualitative
and approximate quantitative model checking algorithms to exist. In this paper,
we explore how to extend the notion of decisiveness to Markov decision
processes. Compared to Markov chains, the extra non-determinism can be resolved
in an adversarial or cooperative way, yielding two natural notions of
decisiveness. We then explore whether these notions yield model checking
procedures concerning the infimum and supremum probabilities of reachability
properties.
