  We consider two-player stochastic games played on a finite graph for
infinitely many rounds. Stochastic games generalize both Markov decision
processes (MDP) by adding an adversary player, and two-player deterministic
games by adding stochasticity. The outcome of the game is a sequence of
distributions over the states of the game graph. We consider synchronizing
objectives, which require the probability mass to accumulate in a set of target
states, either always, once, infinitely often, or always after some point in
the outcome sequence; and the winning modes of sure winning (if the accumulated
probability is equal to 1) and almost-sure winning (if the accumulated
probability is arbitrarily close to 1).
  We present algorithms to compute the set of winning distributions for each of
these synchronizing modes, showing that the corresponding decision problem is
PSPACE-complete for synchronizing once and infinitely often, and PTIME-complete
for synchronizing always and always after some point. These bounds are
remarkably in line with the special case of MDPs, while the algorithmic
solution and proof technique are considerably more involved, even for
deterministic games. This is because those games have a flavour of imperfect
information, in particular they are not determined and randomized strategies
need to be considered, even if there is no stochastic choice in the game graph.
Moreover, in combination with stochasticity in the game graph, finite-memory
strategies are not sufficient in general (for synchronizing infinitely often).
