Dialogue games are a dynamic form of argumentation, with multiple parties pooling their arguments with the intention of settling an issue. Such games can have a variety of structures, and may be collaborative or competitive, depending on the motivations of the parties. Strategic argumentation is a class of competitive dialogue games in which two players take turns in contributing their arguments, each attempting to have an issue settled in the way that they would prefer. Thus strategic argumentation games are less about discovering a joint truth than about a player imposing their view on an opponent. They are reflective of legal argumentation. In the games we study, players have perfect information of the moves players make, but incomplete information on the possible moves (arguments) that other players have available to them. We look both at games using logically structured arguments and games using abstract arguments. We show that playing these games can be computationally hard. We also examine issues of corruption in such games, and discuss approaches to foiling it.