Computational Aspects of Multi-Winner Approval Voting

TL;DR

This paper investigates the computational complexity of multi-winner approval voting rules, establishing NP-hardness results for winner determination and strategic voting scenarios, thereby highlighting computational challenges in these voting systems.

Contribution

It proves NP-hardness of winner computation for proportional approval voting and analyzes the complexity of strategic voting for individual and groups of voters.

Findings

Winner determination for proportional approval voting is NP-hard.

Computing best responses for strategic voters is NP-hard in many settings.

The rules are not strategyproof even with simple preferences.

Abstract

We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show that computing the winner for proportional approval voting is NP-hard, closing a long standing open problem. As none of the rules are strategyproof, even for dichotomous preferences, we study various strategic aspects of the rules. In particular, we examine the computational complexity of computing a best response for both a single agent and a group of agents. In many settings, we show that it is NP-hard for an agent or agents to compute how best to vote given a fixed set of approval ballots from the other agents.