Designing mechanisms for environments without transfers, market designers usually restrict attention to ordinal mechanisms. Ordinal mechanisms are simpler for both designers and participants but miss potentially welfare-relevant information. Under what conditions focusing on ordinal mechanisms is without loss? We show that, in general, all group strategy-proof mechanisms are ordinal. All mechanisms maximizing an Arrovian social welfare function are ordinal; in a large class of environments, such Arrovian efficiency is implied by Pareto efficiency and a simple auditability condition. Strategy-proof mechanisms that are simple to audit are also ordinal. As applications, we characterize important classes of mechanisms in public choice as well as single-unit-demand and multiple-unit-demand allocation of private goods.