Approval-Based Committee Voting under Uncertainty (2407.19391v1)

Abstract: We study approval-based committee voting in which a target number of candidates are selected based on voters' approval preferences over candidates. In contrast to most of the work, we consider the setting where voters express uncertain approval preferences and explore four different types of uncertain approval preference models. For each model, we study the problems such as computing a committee with the highest probability of satisfying axioms such as justified representation.

• The paper's main contribution is the introduction of four distinct uncertainty models to analyze approval-based committee voting under realistic voter conditions.
• It applies rigorous computational complexity analysis, showing NP-completeness for the committee existence problem and #P-completeness for calculating JR probabilities.
• The findings offer practical insights for designing robust decision-making systems and recommendation algorithms in environments with uncertain voter preferences.

Approval-Based Committee Voting under Uncertainty

The paper "Approval-Based Committee Voting under Uncertainty" advances the field of computational social choice by addressing the challenge of selecting committees under uncertain approval preferences from voters. Traditional studies on approval-based committee (ABC) voting generally focus on deterministic voter preferences to achieve justified representation (JR) and other proportional representation axioms. In contrast, the authors of this paper investigate scenarios where the approval data is uncertain, thus reflecting more realistic conditions where precise voter preferences may not be available.

Uncertainty Models and Key Contributions

The authors introduce four distinct models of uncertain approval preferences:

1. Joint Probability Model: This model considers a probability distribution over all possible approval profiles.
2. Lottery Model: In this framework, each voter independently assigns probabilities to their potential approval sets.
3. Candidate-Probability Model: Here, each voter’s approval of each candidate is independently modeled by a probability.
4. Three-Valued Approval (3VA) Model: Each voter categorizes candidates into fully approved, fully disapproved, or uncertain, where uncertain implies an equal probability of approval or disapproval.

These models aim to better handle cases where only partial information about voters’ preferences is available, such as historical voting data or data aggregated from differing voter preferences within a group.

For each uncertainty model, the authors undertake a computational complexity analysis of several significant problems related to JR. These problems include understanding whether a given committee satisfies JR with any positive probability (IsPossJR), with certainty (IsNecJR), and whether such a committee can be constructed (ExistsNecJR). Additionally, the authors explore the likelihood (JR-Probability) and optimization (MaxJR) of forming a committee that meets JR.

Complexity Results and Insights

Among the critical findings, the paper establishes the NP-completeness of the ExistsNecJR across all uncertainty models addressed. Notably, this problem remains NP-complete even under simplified conditions, such as when only two plausible approval profiles exist in the Joint Probability model. Furthermore, JR-Probability is shown to be #P-complete in the 3VA and Candidate Probability models, exemplifying the computational challenges faced when attempting to calculate the likelihood of a committee satisfying JR.

The complexity analysis exposes the nuanced difficulties of manipulating voting systems under preference uncertainty. For instance, the complexity categories for IsPossJR reveal the varying computational demands presented by different models, with the problem being polynomial-time solvable for some models but NP-complete for the Lottery model.

Implications and Future Directions

The theoretical contributions of this paper have practical implications, particularly in the design and evaluation of collective decision-making systems where exact voter preferences cannot be ascertained. Furthermore, the modeling approach can inform developments in recommendation systems and other domains where approval-like interactions are prone to uncertainty.

The paper opens avenues for future research. One potential direction is exploring approximation or fixed-parameter tractable solutions to address the computational challenges identified, especially given the prominence of hard complexity results. Additionally, extending the paper of uncertainty to other vital properties like Pareto optimality and fairness can broaden our understanding and application of fair decision-making in uncertain environments.

In summary, by addressing approval-based committee voting under uncertain preferences with rigor, the paper enriches the existing literature and provides a groundwork for future explorations that align more closely with real-world scenarios in multi-agent decision processes.