  In this short note we study a class of multi-player, turn-based games with
deterministic state transitions and reachability / safety objectives (this
class contains as special cases "classic" two-player reachability and safety
games as well as multi-player "stay--in-a-set" and "reach-a-set" games).
Quantitative and qualitative versions of the objectives are presented and for
both cases we prove the existence of a deterministic and memoryless Nash
equilibrium; the proof is short and simple, using only Fink's classic result
about the existence of Nash equilibria for multi-player discounted stochastic
games.
