The structure of naming systems in natural languages hinges on a trade-off between high informativeness and low complexity. Prior work capitalizes on information theory to formalize these notions; however, these studies generally rely on two simplifications: (i) optimal listeners, and (ii) universal communicative need across languages. Here, we address these limitations by introducing an information-theoretic framework for discrete object naming systems, and we use it to prove that an optimal trade-off is achievable if and only if the listener's decoder is equivalent to the Bayesian decoder of the speaker. Adopting a referential game setup from emergent communication, and focusing on the semantic domain of kinship, we show that our notion of optimality is not only theoretically achievable but also emerges empirically in learned communication systems.