The amount of information conveyed by linguistic conventions depends on their precision, yet the codes that humans and other animals use to communicate are quite ambiguous: they may map several vague meanings to the same symbol. How does semantic precision evolve, and what are the constraints that limit it? We studied the evolution of semantic conventions, in the form of variable mappings between symbols and their referents, in a multiplayer gaming app built around a referential communication game, where a sender had to indicate a colour to a receiver, using only black and white symbols. We expected that the players’ mappings between symbols and colours would grow more specific over time, precision being measured as sense entropy, through a selection process whereby precise mappings are preferentially copied. We found that players become increasingly more precise in their use of symbols over the course of their careers. This trend did not, however, result from selective copying of precise mappings. We explore the implications of this result for the study of lexical ambiguity, Zipf’s Law of Meaning, and disagreements over semantic conventions.