We characterize a class of indirect answers to yes/no questions, alternative answers, where information is given that is not directly asked about, but which might nonetheless address the underlying motivation for the question. We develop a model rooted in game theory that generates these answers via strategic reasoning about possible unobserved domain-level user requirements. We implement the model within an interactive question answering system simulating real estate dialogue. The system learns a prior probability distribution over possible user requirements by analyzing training dialogues, which it uses to make strategic decisions about answer selection. The system generates pragmatically natural and interpretable answers which make for more efficient interactions compared to a baseline.