The bone morphogenetic protein (BMP) signaling pathway comprises multiple ligands and receptors that interact promiscuously with one another and typically appear in combinations. This feature is often explained in terms of redundancy and regulatory flexibility, but it has remained unclear what signal-processing capabilities it provides. Here, we show that the BMP pathway processes multi-ligand inputs using a specific repertoire of computations, including ratiometric sensing, balance detection, and imbalance detection. These computations operate on the relative levels of different ligands and can arise directly from competitive receptor-ligand interactions. Furthermore, cells can select different computations to perform on the same ligand combination through expression of alternative sets of receptor variants. These results provide a direct signal-processing role for promiscuous receptor-ligand interactions and establish operational principles for quantitatively controlling cells with BMP ligands. Similar principles could apply to other promiscuous signaling pathways.