Anonymity and Strategy-Proofness in Public Facility Location with Single-Peaked and Single-Dipped Preferences

Alapfogalmak

This paper characterizes strategy-proof and type-anonymous rules for locating a public facility when agents have either single-peaked or single-dipped preferences, building upon Moulin's median voter rules and quota majority methods.

Kivonat

Bibliographic Information:

Gallo, O. (2024). Anonymity and strategy-proofness on a domain of single-peaked and single-dipped preferences (No. 2410.03387). arXiv.

Research Objective:

This paper aims to characterize strategy-proof and type-anonymous social choice rules for locating a public facility on a line when agents have either single-peaked or single-dipped preferences. The research seeks to identify rules that incentivize truthful preference revelation while ensuring equal decision-making power for all agents, regardless of their preference type.

Methodology:

The paper employs a theoretical and axiomatic approach. It builds on existing literature on strategy-proofness in single-peaked and single-dipped domains, particularly the works of Moulin (1980, 1983) and Alcalde-Unzu et al. (2024). The authors introduce the concept of "type-anonymity" to address the mixed preference domain and analyze the constraints it imposes on strategy-proof rules.

Key Findings:

The paper presents two characterizations of strategy-proof and type-anonymous rules:

1. Two-step procedure: The first step involves computing the median between the peaks of agents with single-peaked preferences and a fixed collection of values (single alternatives or pairs of contiguous alternatives). If the median is a pair, a "double-quota majority method" is applied in the second step to choose between the two alternatives, considering the preferences of both single-peaked and single-dipped agents.
2. Type-anonymous left coalition system: This characterization builds on Alcalde-Unzu et al. (2024) and defines a set of "winning" coalitions for each potential outcome based on the minimal support required. The rule selects the first alternative with sufficient support, considering the number of agents supporting each alternative rather than their identities.

The paper proves the equivalence of these two characterizations.

Main Conclusions:

The research demonstrates that strategy-proof and type-anonymous rules in this mixed preference domain can be characterized by intuitive procedures that combine elements of median voter rules and quota majority methods. The findings contribute to the understanding of mechanism design in settings with heterogeneous preferences.

Significance:

This research is significant for its contribution to social choice theory and mechanism design, particularly in the context of public facility location. The results have implications for designing fair and efficient decision-making mechanisms in situations where individuals may have conflicting preferences due to the nature of the public facility.

Limitations and Future Research:

The paper focuses on a specific mixed preference domain with single-peaked and single-dipped preferences. Future research could explore the implications of relaxing these assumptions or considering other types of restricted preference domains. Additionally, investigating the computational complexity of implementing these rules and their robustness to strategic manipulation by coalitions could be valuable extensions of this work.

Statisztikák

Idézetek

Főbb Kivonatok

Anonymity and strategy-proofness on a domain of single-peaked and single-dipped preferences

by Oihane Gallo : arxiv.org 10-07-2024

https://arxiv.org/pdf/2410.03387.pdf

Mélyebb kérdések

How could the proposed characterizations be extended to accommodate more complex preference domains, such as those involving multiple peaks or dips?

Extending the characterizations to accommodate multiple peaks or dips presents a significant challenge. Here's why and some potential directions:
Challenges:

Loss of Structure: Single-peaked and single-dipped preferences possess a natural "median voter" structure that facilitates strategy-proof mechanisms. Multiple peaks/dips disrupt this, making it harder to find aggregation methods that prevent manipulation.
Increased Complexity: The characterizations rely on the relative positioning of peaks, dips, and phantoms. With multiple peaks/dips, the possible preference orderings become vastly more complex, making it difficult to define analogous concepts like "minimal double-quotas" or "left coalition systems."
Computational Burden: Determining the outcome of even the existing mechanisms (median-based, double-quota majority) becomes computationally more intensive with the increasing complexity of preferences.
Potential Directions:

Restricted Multi-Peaked Domains: Instead of allowing completely arbitrary multiple peaks/dips, explore domains with limited numbers or specific configurations of peaks/dips. For example:

Two Peaks/Dips: Investigate if a three-step mechanism could work, where initial steps narrow down choices and a final step uses a restricted binary choice method.
Semi-Single-Peaked: Allow one peak and multiple dips (or vice-versa), potentially leveraging existing results as building blocks.

Approximation Mechanisms: If exact strategy-proofness is impossible, investigate mechanisms that are "approximately" strategy-proof. These might incentivize truth-telling in most cases or limit the potential gain from manipulation.

Relaxing Anonymity: Explore fairness notions weaker than type-anonymity. This might provide more design flexibility, potentially leading to strategy-proof mechanisms in broader multi-peaked/dipped domains.

Could there be alternative fairness notions beyond type-anonymity that are relevant in this context, and how would they affect the characterization of strategy-proof rules?

Yes, several alternative fairness notions could be considered:

Geographical Proximity: Instead of just treating groups anonymously, prioritize outcomes that minimize the average or maximum distance between agents and the facility. This is particularly relevant for essential services.

Impact on Characterization: This would likely necessitate distance-based mechanisms, potentially leading to trade-offs between strategy-proofness and optimizing proximity.

Vulnerable Group Considerations: Assign different weights or priorities to specific groups (e.g., elderly, disabled) to ensure their needs are met.

Impact on Characterization: This might involve weighted voting schemes or adjustments to quotas in the double-quota majority method, potentially making the characterization more complex.

Envy-Freeness: Aim for outcomes where no agent envies the location allocated to another agent. This is challenging in facility location as preferences are inherently location-dependent.

Impact on Characterization: Achieving envy-freeness might be highly restrictive, potentially leading to a very limited set of strategy-proof and envy-free rules, or even impossibility results.

Minimizing Disenfranchisement: Design mechanisms where the outcome cannot be too far from the ideal point of any agent, preventing extreme dissatisfaction.

Impact on Characterization: This might involve range restrictions on the possible outcomes or incorporating a "veto" mechanism, potentially complicating the characterization.

Incorporating these fairness notions would likely lead to more complex characterizations, potentially requiring new axioms and mathematical tools to capture the interplay between strategy-proofness and the specific fairness criteria.

What are the practical implications of these findings for real-world public facility location decisions, considering the challenges of preference elicitation and aggregation?

While the theoretical results provide valuable insights, translating them into practice for public facility location decisions faces several challenges:

Preference Elicitation:

Complexity: Real-world preferences are often nuanced and may not neatly fit into single-peaked/dipped categories. Eliciting complex preferences accurately is difficult.
Strategic Behavior: Even if preferences are well-defined, individuals might misreport them strategically, especially if they understand the mechanism being used.

Preference Aggregation:

Computational Limits: As the number of agents and alternatives increases, the computational burden of determining the outcome of even relatively simple mechanisms can become significant.
Trade-offs: Balancing strategy-proofness with other desirable properties like fairness (as discussed above) often involves trade-offs. Finding acceptable compromises can be politically challenging.

Communication and Transparency:

Explaining the Mechanism: Public acceptance of a decision often hinges on understanding how it was made. Explaining complex, strategy-proof mechanisms to the public in a clear and understandable way is crucial but difficult.
Building Trust: Lack of transparency in preference aggregation can erode public trust. Mechanisms should be designed to be auditable and justifiable.

Practical Considerations:

Hybrid Approaches: Combine aspects of strategy-proof mechanisms with other methods, such as public consultations or multi-criteria decision analysis, to incorporate broader considerations.
Approximate Solutions: Instead of seeking perfect strategy-proofness, focus on mechanisms that are computationally tractable and minimize the potential for manipulation in practice.
Iterative Processes: Use a staged approach, starting with simpler mechanisms to narrow down options and then employing more sophisticated methods for final decisions.
Key Takeaway:
While achieving perfect strategy-proofness and fairness in complex real-world scenarios is challenging, these theoretical findings provide a framework for designing more robust and transparent public facility location processes. By carefully considering the trade-offs and practical limitations, decision-makers can strive for mechanisms that balance competing objectives and foster public trust.