We propose to study concurrent games on a new extension of Vector Addition Systems with States, where inhibition conditions are added for modeling purposes. Games are a well-suited framework to solve control problems, and concurrent semantics reflect realistic situations where the environment can always produce a move before the controller, although it is never required to do so. This is in contrast with previous works, which focused mainly on turn-based semantics. Moreover, we consider asymmetric games, where environment and controller do not have the same capabilities, although they both have restricted power. In this setting, we investigate reachability and safety objectives, which are not dual to each other anymore, and we prove that (i) reachability games are undecidable for finite targets, (ii) they are 2-EXPTIME-complete for upward-closed targets and (iii) safety games are co-NP-complete for finite, upward-closed and semi-linear targets. Moreover, for the decidable cases, we build a finite representation of the corresponding controllers.