Stable Matching with Ties: How Good Can Algorithmic Approximations Be?

Core Concepts

While stable matchings are not unique in markets where participants can have indifference between options, randomized algorithms can guarantee a logarithmic fraction of a participant's best possible outcome.

Abstract

• Bibliographic Information: Lin, S., Mauras, S., Merlis, N., & Perchet, V. (2024). Stable Matching with Ties: Approximation Ratios and Learning. arXiv:2411.03270v1 [cs.GT].
• Research Objective: This paper investigates the limits of achieving fairness in stable matching scenarios where one side of the market (e.g., workers) might have ties in their preferences (e.g., indifference between jobs). The authors aim to determine how much of their optimal stable share (OSS) - the best they could achieve in any stable matching - can be guaranteed to each worker using randomized algorithms.
• Methodology: The authors utilize theoretical analysis and algorithm design to establish lower and upper bounds on the achievable OSS ratio. They first demonstrate the limitations of using only stable matchings by proving a lower bound that scales linearly with the number of workers. Then, they propose an algorithm that constructs a distribution over internally stable matchings, achieving a logarithmic approximation of the OSS. This algorithm is further analyzed for its robustness to uncertainty in the utility function and adapted for online learning settings using a multi-armed bandit framework.
• Key Findings:
  • Distributions over stable matchings alone cannot guarantee a good OSS ratio.
  • Randomizing over all matchings leads to a logarithmic lower bound on the OSS ratio.
  • The proposed algorithm, which randomizes over internally stable matchings, achieves a logarithmic OSS ratio, proving its asymptotic optimality.
  • The algorithm is robust to uncertainty in utility estimation and can be adapted for online learning, achieving sublinear regret in many cases.
• Main Conclusions: This work provides a novel perspective on fairness in stable matching with ties by introducing the concept of the OSS ratio. The proposed algorithm offers a practical and near-optimal solution for achieving fairness in such scenarios. Furthermore, the extension to uncertain utilities and online learning makes the approach applicable to a wide range of real-world problems.
• Significance: This research significantly contributes to the field of stable matching by addressing the challenge of ties in preferences and providing an efficient and robust solution. The findings have implications for various applications, including labor markets, online platforms, and resource allocation problems.
• Limitations and Future Research: While the paper provides a comprehensive analysis, future research could explore alternative fairness notions and investigate the trade-off between fairness and other desirable properties in stable matching with ties.

Stats

There are N workers and K jobs.
Each worker can perform one job and each job requires one worker.
The utility of worker w for performing job a is represented as U(w, a) ∈ [0, 1].
Each job a has a strict skill ordering over workers (≻a).
The optimal stable share (OSS) of a worker is the highest utility they can receive in any stable matching.

Quotes

Key Insights Distilled From

Stable Matching with Ties: Approximation Ratios and Learning

by Shiyun Lin, ... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03270.pdf

Deeper Inquiries

How could the concept of the OSS ratio be extended to many-to-one matching scenarios, where one side of the market can be matched to multiple entities on the other side?

Extending the OSS ratio to many-to-one matching scenarios, such as college admissions where a college can accept multiple students, requires careful consideration of the matching structure and the definition of stability. Here's a possible approach:
1. Redefining Stability:

Weak Stability: A matching is weakly stable if no student-college pair exists where the student prefers the college over their assigned colleges and the college has a vacant slot or prefers the student over one of their admitted students.
Quota Constraints: Colleges would have quotas indicating the maximum number of students they can accept.
2. Adapting OSS:

OSS for Students: The OSS for a student remains largely the same – the maximum total utility achievable in any stable matching, summing the utilities of all assigned colleges.
OSS for Colleges: Defining OSS for colleges is trickier. One approach could be:

Average OSS: The average OSS across all students the college is matched with in a stable matching. This promotes overall student satisfaction.
Minimum OSS: The minimum OSS of any student matched to the college in a stable matching. This focuses on a worst-case fairness guarantee for students.
3. OSS Ratio:

Student-Oriented: The OSS ratio for students would be the ratio of their achieved utility to their OSS, similar to the one-to-one case.
College-Oriented: The OSS ratio for colleges would depend on the chosen OSS definition (average or minimum).
Challenges and Considerations:

Computational Complexity: Finding stable matchings in many-to-one scenarios with ties is generally more complex than in one-to-one settings.
Fairness Trade-offs: Different OSS definitions for colleges lead to different fairness implications. Average OSS might benefit colleges with high overall demand, while minimum OSS might favor colleges with less competitive admissions.
Strategic Behavior: The presence of quotas and multiple assignments might create new opportunities for strategic manipulation by both students and colleges.

Could there be alternative fairness metrics in stable matching with ties that prioritize different aspects of fairness, such as minimizing the maximum envy between workers?

Yes, alternative fairness metrics can be designed for stable matching with ties, focusing on aspects beyond the OSS ratio. Minimizing envy is a prime example:
1. Envy-Based Metrics:

Maximum Envy: This metric measures the largest utility difference between a worker and another worker they envy (i.e., a worker assigned a job they prefer). Minimizing maximum envy promotes a sense of fairness where no worker feels significantly disadvantaged compared to their peers.
Total Envy: This metric sums up the envy experienced by all workers in the matching. Minimizing total envy aims to reduce overall dissatisfaction stemming from envy.
2. Other Fairness Considerations:

Prioritizing Disadvantaged Workers: Metrics could be designed to prioritize the utility of workers belonging to historically disadvantaged groups, ensuring they are not disproportionately assigned less desirable jobs.
Balancing Utility and Stability: Hybrid metrics could combine aspects of stability and fairness, seeking matchings that are both stable and optimize a fairness objective. For example, finding a stable matching with the minimum maximum envy.
Challenges and Trade-offs:

Computational Tractability: Optimizing for some fairness metrics, like minimizing maximum envy, might be computationally hard. Approximation algorithms or heuristics might be necessary.
Conflicting Objectives: Fairness goals can sometimes conflict. For instance, minimizing maximum envy might not always lead to the most efficient allocation of jobs based on worker skills.
Defining Envy with Ties: Carefully defining envy in the presence of ties is crucial. Should a worker envy another worker assigned a job they are indifferent to?

How can the insights from this research be applied to design mechanisms for fair and efficient resource allocation in dynamic environments, such as online platforms with constantly changing user preferences?

The insights from stable matching with ties and the OSS ratio have valuable implications for designing fair and efficient resource allocation mechanisms in dynamic environments like online platforms:
1. Dynamic OSS Approximation:

Periodic Re-matching: The algorithm for approximating the OSS ratio could be run periodically to adapt to changing preferences. This ensures that over time, users receive allocations that reflect their evolving preferences.
Bandit Learning: Incorporate bandit learning techniques to handle the exploration-exploitation dilemma. The platform can learn user preferences over time by observing their responses to different allocations.
2. Handling Uncertainty and Fluctuations:

Epsilon-Stability: The concept of ϵ-stability provides robustness to minor fluctuations in preferences. The platform can tolerate small instabilities to avoid frequent re-matching and ensure a smoother user experience.
Preference Elicitation: Implement mechanisms for users to update their preferences explicitly, allowing for more responsive adjustments to the allocation mechanism.
3. Fairness in Dynamic Settings:

Temporal Fairness: Consider fairness not just in a single snapshot but over time. Users who might not get their most preferred allocation initially should have a fair chance to receive it later as preferences change.
Fairness Beyond OSS: Explore incorporating other fairness metrics, such as envy minimization or prioritizing specific user groups, to address fairness concerns in the platform's specific context.
Challenges in Dynamic Environments:

Real-time Adaptation: Balancing responsiveness to changing preferences with the stability of allocations is crucial. Frequent re-matching can be disruptive, while infrequent updates might lead to dissatisfaction.
Scalability: Algorithms need to be scalable to handle a large number of users and resources, especially in platforms with millions of participants.
Strategic Behavior: Users might act strategically to game the system, especially if they understand how the allocation mechanism adapts to their preferences. Designing mechanisms robust to manipulation is essential.