This paper studies welfare maximizing allocation of indivisible objects to ex-ante identical agents in the absence of monetary transfers. The agents, each with a unit demand, share a common ranking of the objects, and are privately informed about their own valuations. The structure of the optimal allocation policy depends on the agents' relative valuation of the objects and the variation of this relative valuation across different types. When this variation is small, the required loss of welfare for eliciting agents' private information exceeds its benefits. In this case, evenly randomized allocation is optimal. When this variation is significantly large, it is optimal to waste the less preferred object-not always allocate it to agents-to provide necessary incentives for information elicitation. The planner then uses this information to increase the frequency of allocating the more preferred object to the agent favored by the first best policy. Regardless of the size of the variation, it is never optimal to waste the more preferred object. We also propose an exchange game that implements the incentive efficient allocation. (C) 2020 Elsevier B.V. All rights reserved.