The Complexity of Learning Approval-Based Multiwinner Voting Rules

TL;DR

This paper investigates the learnability of approval-based multiwinner voting rules, showing that while they can be learned efficiently from samples, determining specific winning committees remains computationally hard.

Contribution

It demonstrates that ABCS voting rules are PAC-learnable with polynomial samples, but deciding committee winners is computationally intractable.

Findings

Sample complexity for learning ABCS rules is polynomial.

Deciding if a committee can be winning under an ABCS rule is NP-hard.

Results extend to sequential Thiele rules.

Abstract

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. According to ABCS rules, each committee of $k$ candidates collects from each voter a score, which 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 rule 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 recently due to their simplicity.