  We consider discrete-time Markov decision processes in which the decision
maker is interested in long but finite horizons. First we consider reachability
objective: the decision maker's goal is to reach a specific target state with
the highest possible probability. Formally, strategy $\sigma$ overtakes another
strategy $\sigma'$, if the probability of reaching the target state within
horizon $t$ is larger under $\sigma$ than under $\sigma'$, for all sufficiently
large $t\in\NN$. We prove that there exists a pure stationary strategy that is
not overtaken by any pure strategy nor by any stationary strategy, under some
condition on the transition structure and respectively under genericity. A
strategy that is not overtaken by any other strategy, called an overtaking
optimal strategy, does not always exist. We provide sufficient conditions for
its existence.
  Next we consider safety objective: the decision maker's goal is to avoid a
specific state with the highest possible probability. We argue that the results
proven for reachability objective extend to this model. We finally discuss
extensions of our results to two-player zero-sum perfect information games.
