The Complexity of Learning Approval-Based Multiwinner Voting Rules

We study the PAC learnability of multiwinner voting, focusing on the class of approval-based committee scoring (ABCS) rules. These are voting rules applied on profiles with approval ballots, where each voter approves some of the candidates. ABCS rules adapt positional scoring rules in single-winner voting by assuming that each committee of $k$ candidates collects from each voter a score, that depends on the size of the voter's ballot and on the size of its intersection with the committee. Then, committees of maximum score are the winning ones. Our goal is to learn a target rule (i.e., to learn the corresponding scoring function) using information about the winning committees of a small number of sampled profiles. Despite the existence of exponentially many outcomes compared to single-winner elections, we show that the sample complexity is still low: a polynomial number of samples carries enough information for learning the target committee with high confidence and accuracy. Unfortunately, even simple tasks that need to be solved for learning from these samples are intractable. We prove that deciding whether there exists some ABCS rule that makes a given committee winning in a given profile is a computationally hard problem. Our results extend to the class of sequential Thiele rules, which have received attention due to their simplicity.