Objects of different quality are to be allocated to agents. Agents can receive at most one object, and there are not enough high-quality objects for every agent.The value to the social planner from allocating objects to any given agent depends on that agent’s private information. The social planner is unable to use transfers to give incentives for agents to convey their private information. Instead, she is able to imperfectly verify their reports through signals that are positively affiliated with each agent’s type. We characterize mechanisms that maximize the social planner’s expected payoff. In the optimal mechanism, each agent chooses one of various tracks, which are characterized by two thresholds. If the agent’s signal exceeds the upper threshold of the chosen track, the agent receives a high-quality object, if it is between the two thresholds, he receives a low-quality object, and if it is below the lower threshold, he receives no object.