Algorithms are widely used to guide high-stakes decisions, from medical recommendations to loan approvals. Designers are increasingly optimizing not only for accuracy but also "fairness" i.e. how much accuracy varies across different subgroups. We define and characterize a fairness-accuracy frontier, consisting of the optimal points across a broad range of criteria for trading off fairness and accuracy. Our results identify how the algorithm's inputs govern the shape of this frontier, showing (for example) that fairness considerations matter for the frontier precisely when the inputs fail a condition we call group-balance. We next study an information design problem where the designer controls inputs (e.g., by legally banning an input) but the algorithm itself is chosen by another agent. We show that all designers strictly prefer to allow access to group identity if and only if the other inputs satisfy group-balance. (joint with Jay Lu and Xiaosheng Mu)