Multiwinner Analogues of Plurality Rule: Axiomatic and Algorithmic Perspectives

TL;DR

This paper characterizes multiwinner committee scoring rules satisfying the fixed-majority criterion, introduces top-k-counting rules, and analyzes their computational complexity, providing both NP-hardness results and efficient algorithms.

Contribution

It defines and characterizes fixed-majority consistent multiwinner rules as a subclass of top-k-counting rules, and analyzes their computational complexity and algorithms.

Findings

Most rules are NP-hard to compute

Some rules admit polynomial-time algorithms

Exact FPT and approximation algorithms are provided

Abstract

We characterize the class of committee scoring rules that satisfy the fixed-majority criterion. In some sense, the committee scoring rules in this class are multiwinner analogues of the single-winner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-$k$-counting committee scoring rules and show that the fixed majority consistent rules are a subclass of the top-$k$-counting rules. We give necessary and sufficient conditions for a top-$k$-counting rule to satisfy the fixed-majority criterion. We find that, for most of the rules in our new class, the complexity of winner determination is high (that is, the problem of computing the winners is NP-hard), but we also show examples of rules with polynomial-time winner determination procedures. For some of the computationally hard rules, we provide either exact FPT algorithms or approximate polynomial-time algorithms.