  Two-player zero-sum games are a well-established model for synthesising
controllers that optimise some performance criterion. In such games one player
represents the controller, while the other describes the (adversarial)
environment, and controller synthesis corresponds to computing the optimal
strategies of the controller for a given criterion. Asarin and Maler initiated
the study of quantitative games on (non-probabilistic) timed automata by
synthesising controllers which optimise the time to reach a final state. The
correctness and termination of their approach was dependent on exploiting the
properties of a special class of functions, called simple functions, that can
be finitely represented. In this paper we consider quantitative games over
probabilistic timed automata. Since the concept of simple functions is not
sufficient to solve games in this setting, we generalise simple functions to
so-called quasi-simple functions. Then, using this class of functions, we
demonstrate that the problem of solving games with either expected
reachability-time or expected discounted-time criteria on probabilistic timed
automata are in NEXPTIME $\cap$ co-NEXPTIME.
