Language and coordination games

I formalize the role of the self-signaling condition in guaranteeing coordination for pre-play cheap talk games with one-sided communication. I model a preexisting common language by assuming that the Receiver either ignores or follows cheap talk recommendations, but never inverts them. This assumption creates asymmetry between messages, which captures the essence of a common language. It does not rule out any outcome at hand in that every equilibrium outcome of the original game remains an equilibrium outcome in this transformed game. However, applying iterative admissibility to the transformed game yields sharp predictions. If the stage game satisfies a certain self-signaling condition, then the Sender gets her Stackelberg payoff in every iteratively admissible outcome. On the other hand, if the stage game violates a weaker self-signaling condition, miscoordination can happen in an iteratively admissible outcome.