Researchers are using deep learning models to explore the emergence of language in various language games, where agents interact and develop an emergent language to solve tasks. We focus on the factors that determine the expressivity of emergent languages, which reflects the amount of information about input spaces those languages are capable of encoding. We measure the expressivity of emergent languages based on the generalisation performance across different games, and demonstrate that the expressivity of emergent languages is a trade-off between the complexity and unpredictability of the context those languages emerged from. Another contribution of this work is the discovery of message type collapse, i.e. the number of unique messages is lower than that of inputs. We also show that using the contrastive loss proposed by Chen et al. (2020) can alleviate this problem.