Investigating cooperativity of interlocutors is central in studying pragmatics of dialogue. Models of conversation that only assume cooperative agents fail to explain the dynamics of strategic conversations. Thus, we investigate the ability of agents to identify non-cooperative interlocutors while completing a concurrent visual-dialogue task. Within this novel setting, we study the optimality of communication strategies for achieving this multi-task objective. We use the tools of learning theory to develop a theoretical model for identifying non-cooperative interlocutors and apply this theory to analyze different communication strategies. We also introduce a corpus of non-cooperative conversations about images in the GuessWhat?! dataset proposed by De Vries et al. (2017). We use reinforcement learning to implement multiple communication strategies in this context and find that empirical results validate our theory.