Goods and services—public housing, medical appointments, schools—are often allocated to individuals who rank them similarly but differ in their preference intensities. We characterize optimal allocation rules when individual preferences are known and when they are not. Several insights emerge. First-best allocations may involve assigning some agents “lotteries” between high- and low-ranked goods. When preference intensities are private information, second-best allocations always involve such lotteries and, crucially, may coincide with first-best allocations. Furthermore, second-best allocations may entail disposal of services. We discuss a market-based alternative and show how it differs.