Language models pre-trained on large self-supervised corpora, followed by task-specific fine-tuning has become the dominant paradigm in NLP. These pre-training datasets often have a one-to-many structure--e.g. in dialogue there are many valid responses for a given context. However, only some of these responses will be desirable in our downstream task. This raises the question of how we should train the model such that it can emulate the desirable behaviours, but not the undesirable ones. Current approaches train in a one-to-one setup--only a single target response is given for a single dialogue context--leading to models only learning to predict the average response, while ignoring the full range of possible responses. Using text-based games as a testbed, our approach, PASA, uses discrete latent variables to capture the range of different behaviours represented in our larger pre-training dataset. We then use knowledge distillation to distil the posterior probability distribution into a student model. This probability distribution is far richer than learning from only the hard targets of the dataset, and thus allows the student model to benefit from the richer range of actions the teacher model has learned. Results show up to 49% empirical improvement over the previous state-of-the-art model on the Jericho Walkthroughs dataset.