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Language structure in the n-object naming game

We examine a naming game with two agents trying to establish a common vocabulary for n objects. Such efforts lead to the emergence of language that allows for an efficient communication and exhibits some degree of homonymy and synonymy. Although homonymy reduces the communication efficiency, it seems to be a dynamical trap that persists for a long, and perhaps indefinite, time. On the other hand, synonymy does not reduce the efficiency of communication, but appears to be only a transient feature of the language. Thus, in our model the role of synonymy decreases and in the long-time limit it becomes negligible. A similar rareness of synonymy is observed in present natural languages. The role of noise, that distorts the communicated words, is also examined. Although, in general, the noise reduces the communication efficiency, it also regroups the words so that they are more evenly distributed within the available "verbal" space.