  Stochastic timed games (STGs), introduced by Bouyer and Forejt, naturally
generalize both continuous-time Markov chains and timed automata by providing a
partition of the locations between those controlled by two players (Player Box
and Player Diamond) with competing objectives and those governed by stochastic
laws. Depending on the number of players---$2$, $1$, or $0$---subclasses of
stochastic timed games are often classified as $2\frac{1}{2}$-player,
$1\frac{1}{2}$-player, and $\frac{1}{2}$-player games where the $\frac{1}{2}$
symbolizes the presence of the stochastic "nature" player. For STGs with
reachability objectives it is known that $1\frac{1}{2}$-player one-clock STGs
are decidable for qualitative objectives, and that $2\frac{1}{2}$-player
three-clock STGs are undecidable for quantitative reachability objectives. This
paper further refines the gap in this decidability spectrum. We show that
quantitative reachability objectives are already undecidable for $1\frac{1}{2}$
player four-clock STGs, and even under the time-bounded restriction for
$2\frac{1}{2}$-player five-clock STGs. We also obtain a class of
$1\frac{1}{2}$, $2\frac{1}{2}$ player STGs for which the quantitative
reachability problem is decidable.
