This paper presents MDP policy learning for agents to learn strategic behavior–how to play board games–during multimodal dialogues. Policies are trained offline in simulation, with dialogues carried out in a formal language. The agent has a temporary belief state for the dialogue, and a persistent knowledge store represented as an extensive-form game tree. How well the agent learns a new game from a dialogue with a simulated partner is evaluated by how well it plays the game, given its dialogue-final knowledge state. During policy training, we control for the simulated dialogue partner’s level of informativeness in responding to questions. The agent learns best when its trained policy matches the current dialogue partner’s informativeness. We also present a novel data collection for training natural language modules. Human subjects who engaged in dialogues with a baseline system rated the system’s language skills as above average. Further, results confirm that human dialogue partners also vary in their informativeness.