Models of conversation that rely on a strong notion of cooperation don’t apply to strategic conversation — that is, to conversation where the agents’ motives don’t align, such as courtroom cross examination and political debate. We provide a game-theoretic framework that provides an analysis of both cooperative and strategic conversation. Our analysis features a new notion of safety that applies to implicatures: an implicature is safe when it can be reliably treated as a matter of public record. We explore the safety of implicatures within cooperative and non cooperative settings. We then provide a symbolic model enabling us (i) to prove a correspondence result between a characterisation of conversation in terms of an alignment of players’ preferences and one where Gricean principles of cooperative conversation like Sincerity hold, and (ii) to show when an implicature is safe and when it is not.