Extensions and Vagueness of Language under Two-Dimensional State Uncertainty

We examine languages under two-dimensional uncertainty about the stateÑ the type and support of the state-distribution as well as the state is ex-ante uncertain. We consider communication between an uninformed decision maker and an imperfectly informed speaker. We show that this two-dimensional uncertainty leads the extension of language to be vague in the sense that the speaker with different knowledge of the state-distribution uses the same symbol for the different extent of information on the state. We also explore languages when the players contract over the number of symbols due to bounded rationality and when they possesses different prior beliefs.