ABSTRACTIn environments without transfers, market designers usually restrict attention to ordinal mechanisms. Ordinal mechanisms are simpler but miss potentially welfare‐relevant information. Under what conditions is it without loss to focus on ordinal mechanisms? We show that all group strategy‐proof mechanisms are ordinal. While not all Pareto efficient mechanisms are ordinal, all mechanisms maximizing an Arrovian social welfare function must be. These results accommodate feasibility constraints and allow for simple or complex demand. As applications, we characterize important classes of mechanisms in public choice and in the allocation of private goods under unit and multiunit demand.