The spontaneous exchange of turns is a central aspect of human communication. Although turn-taking conventions come to us naturally, artificial dialogue agents struggle to coordinate, and must rely on hard-coded rules to engage in interactive conversations with human interlocutors. In this paper, we investigate the conditions under which artificial agents may naturally develop turn-taking conventions in a simple language game. We describe a cooperative task where success is contingent on the exchange of information along a shared communication channel where talking over each other hinders communication. Despite these environmental constraints, neural-network based agents trained to solve this task with reinforcement learning do not systematically adopt turn-taking conventions. However, we find that agents that do agree on turn-taking protocols end up performing better. Moreover, agents that are forced to perform turn-taking can learn to solve the task more quickly. This suggests that turn-taking may help to generate conversations that are easier for speakers to interpret.