Computational Aspects of Multi-Winner Approval Voting (1407.3247v1)

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.

• The paper analyzes the computational complexity of Satisfaction, Proportional, and Reweighted Approval Voting rules, proving that computing winners under Proportional Approval Voting is NP-hard.
• All studied rules are susceptible to strategic manipulation, and computing a beneficial strategic vote is NP-hard, raising concerns about election integrity.
• The computational complexities, particularly NP-hardness for PAV, suggest potential barriers to practical implementation and highlight needs for optimization and strategyproof mechanisms.

Computational Complexity in Multi-Winner Approval Voting

The paper, "Computational Aspects of Multi-Winner Approval Voting," authored by Haris Aziz et al., explores the computational intricacies associated with three specific approval-based voting rules aimed at selecting multiple winners. These rules include Satisfaction Approval Voting, Proportional Approval Voting, and Reweighted Approval Voting. The paper presents novel insights, notably establishing the NP-hard status of computing winners in Proportional Approval Voting, thereby resolving an enduring open question within computational social choice.

Overview of Approval-Based Multi-Winner Voting Rules

Approval voting is a versatile mechanism allowing voters to endorse multiple candidates, and this flexibility becomes crucial in contexts requiring the election of multiple winners, such as parliamentary elections, faculty hiring, and recommendation systems. The paper critiques the transition of approval voting from a single-winner to a multi-winner framework, highlighting challenges in preserving fairness and proportional representation in such generalizations.

1. Satisfaction Approval Voting (SAV) looks to maximize agent satisfaction, calculated as the fraction of an agent's approved candidates that get elected.
2. Proportional Approval Voting (PAV) addresses diminishing returns by weighting approvals based on the number of elected candidates from an agent's ballot, with complexity arising in winner determination due to its NP-hard status.
3. Reweighted Approval Voting (RAV) employs a sequential approach where the weight of approvals is adjusted after each round of candidate selection.

Computational Complexity Findings

The core contribution of the paper is its exploration of computational complexities related to these voting rules, particularly under strategic voting scenarios. The authors confirm that computing winners under PAV is NP-hard even if agents approve only two candidates each. Furthermore, the paper reveals that all studied rules are susceptible to strategic manipulation, and computing a beneficial strategic vote is NP-hard, which underscores the potential obstacles in algorithmic implementation and election integrity:

• Winner Manipulation (WM), along with Winning Set Manipulation (WSM), presents computational challenges. For SAV, WSM is NP-hard while its manipulation remains polynomial in simpler cases. PAV, inherently complex in WD, reflects this complexity in manipulation, driving the point that strategy proofing remains an unachieved goal for proportional voting systems.
• Strategic Voting: Despite dichotomous preferences simplifying the utility landscape, all studied rules, barring single-winner approval voting, permit strategic voting, inviting questions about their robustness against tactical voting manipulation.

Implications and Future Directions

The computational complexities identified for PAV, SAV, and RAV raise pertinent questions about their practical applicability in real-world election scenarios. The NP-hardness of core decision processes suggests that computational barriers might limit their deployment in contexts requiring quick and efficient result processing.

Theoretical implications extend beyond algorithm efficiency into the axiomatic evaluation and voter fairness. With computational difficulty becoming a fundamental property of PAV and related rules, future work may span optimization techniques, approximation algorithms, and experimental validations to balance computational feasibility with fair representation objectives.

Moreover, enhancing strategyproof characteristics of these systems can offer significant advancements in electoral practice, demanding interdisciplinary approaches combining game theory, AI, and algorithm research.

Conclusion

This paper provides a comprehensive computational analysis of significant approval-based multi-winner voting rules, offering pivotal advancements in understanding their complexities and strategic vulnerabilities. Beyond the immediate mathematical insights, the implications call for broader explorations into ensuring equitable and computationally tractable election frameworks amidst growing demands for democratic decision-making processes. As the landscape of computational social choice continues to evolve, these investigations offer foundational steps toward more robust multi-winner electoral systems.