Abstract: Most approaches to Open-Domain Question Answering consist of a light-weight retriever that selects a set of candidate passages, and a computationally expensive reader that examines the passages to identify the correct answer. Previous works have shown that as the number of retrieved passages increases, so does the performance of the reader. However, they assume all retrieved passages are of equal importance and allocate the same amount of computation to them, leading to a substantial increase in computational cost. To reduce this cost, we propose the use of adaptive computation to control the computational budget allocated for the passages to be read. We first introduce a technique operating on individual passages in isolation which relies on anytime prediction and a per-layer estimation of early exit probability. We then introduce SKYLINEBUILDER, an approach for dynamically deciding on which passage to allocate computation at each step, based on a resource allocation policy trained via reinforcement learning. Our results on SQuAD-Open show that adaptive computation with global prioritisation improves over several strong static and adaptive methods, leading to a 4.3x reduction in computation while retaining 95% performance of the full model.