  We consider concurrent games played on graphs. At every round of the game,
each player simultaneously and independently selects a move; the moves jointly
determine the transition to a successor state. Two basic objectives are the
safety objective: ``stay forever in a set F of states'', and its dual, the
reachability objective, ``reach a set R of states''. We present in this paper a
strategy improvement algorithm for computing the value of a concurrent safety
game, that is, the maximal probability with which player 1 can enforce the
safety objective. The algorithm yields a sequence of player-1 strategies which
ensure probabilities of winning that converge monotonically to the value of the
safety game.
  The significance of the result is twofold. First, while strategy improvement
algorithms were known for Markov decision processes and turn-based games, as
well as for concurrent reachability games, this is the first strategy
improvement algorithm for concurrent safety games. Second, and most
importantly, the improvement algorithm provides a way to approximate the value
of a concurrent safety game from below (the known value-iteration algorithms
approximate the value from above). Thus, when used together with
value-iteration algorithms, or with strategy improvement algorithms for
reachability games, our algorithm leads to the first practical algorithm for
computing converging upper and lower bounds for the value of reachability and
safety games.
