We study Prioritized Selection Problems in which an organization is presented with a set of individuals, and must choose which subset to accept. The organization makes a selection based on a priority ranking of individuals as well as other observable characteristics. We study outcome based selection rules, which are defined by a collection of feasible selections and a greedy processing algorithm.

We consider a model with k identical tickets. The set of agents (N) is partitioned into a set of groups, and agents have dichotomous preferences: an agent is successful if and only if members of her group receive enough tickets for everyone in the group. We treat the group structure as private information, unknown to the designer. Because there are only k tickets, there can be at most k successful agents. We define the efficiency of a lottery allocation to be the expected number of successful agents, divided by k. If this is at least β, then the allocation is β-efficient. A lottery allocation is fair if each agent has the same success probability, and β-fair if for any pair of agents, the ratio of their success probabilities is at least β.