Multiwinner Voting with Possibly Unavailable Candidates

Selecting a committee that meets diversity and proportionality criteria is a challenging endeavor that has been studied extensively in recent years. This task becomes even more challenging when some of the selected candidates decline the invitation to join the committee. Since the unavailability of one candidate may impact the rest of the selection, inviting all candidates at the same time may lead to a suboptimal committee. Instead, invitations should be sequential and conditional on which candidates invited so far accepted the invitation: the solution to the committee selection problem is a query policy. If invitation queries are binding, they should be safe: one should not query a candidate without being sure that whatever the set of available candidates possible at that stage, her inclusion will not jeopardize committee optimality. Assuming approval-based inputs, we characterize the set of rules for which a safe query exists at every stage. In order to parallelize the invitation process, we investigate the computation of safe parallel queries, and show that it is often hard. We also study the existence of safe parallel queries with respect to proportionality axioms such as extended justified representation.