[Paper Review] Improving Welfare in One-sided Matching using Simple Threshold Queries

Thomas Ma, Vijay Menon+1 more|arXiv (Cornell University)|Nov 27, 2020

Game Theory and Voting Systems36 references4 citations

TL;DR

This paper proposes using simple threshold queries—asking agents if they value an object above a specific utility threshold—to improve welfare in one-sided matching without requiring full cardinal utility elicitation. By leveraging just one bit of extra information per object, the authors design adaptive and non-adaptive algorithms that achieve a welfare approximation factor of Ω(√n) relative to the optimal matching satisfying economic efficiency properties like Pareto optimality or rank-maximality.

ABSTRACT

We study one-sided matching problems where $n$ agents have preferences over $m$ objects and each of them need to be assigned to at most one object. Most work on such problems assume that the agents only have ordinal preferences and usually the goal in them is to compute a matching that satisfies some notion of economic efficiency. However, in reality, agents may have some preference intensities or cardinal utilities that, e.g., indicate that they like an object much more than another object, and not taking these into account can result in a loss in welfare. While one way to potentially account for these is to directly ask the agents for this information, such an elicitation process is cognitively demanding. Therefore, we focus on learning more about their cardinal preferences using simple threshold queries which ask an agent if they value an object greater than a certain value, and use this in turn to come up with algorithms that produce a matching that, for a particular economic notion $X$, satisfies $X$ and also achieves a good approximation to the optimal welfare among all matchings that satisfy $X$. We focus on several notions of economic efficiency, and look at both adaptive and non-adaptive algorithms. Overall, our results show how one can improve welfare by even non-adaptively asking the agents for just one bit of extra information per object.

Motivation & Objective

• Address the welfare loss in one-sided matching when only ordinal preferences are used, despite agents having underlying cardinal utility intensities.
• Overcome the cognitive burden of directly eliciting cardinal utilities by introducing a lightweight query mechanism: threshold queries asking if an agent values an object above a given utility level.
• Design algorithms that guarantee a matching satisfying a desired economic efficiency property (e.g., Pareto optimality, rank-maximality) while approximating the optimal welfare among such matchings.
• Evaluate both adaptive and non-adaptive query strategies to determine the trade-offs between query efficiency and welfare approximation quality.
• Demonstrate that even a minimal amount of cardinal information—one bit per object—can significantly improve social welfare in matching outcomes.

Proposed method

• Use threshold queries of the form 'Do you value object o at least at level v?' to elicit partial cardinal preference information from agents.
• Design adaptive algorithms that dynamically select thresholds based on prior query responses to refine utility estimates and improve matching quality.
• Develop non-adaptive algorithms that predefine a fixed set of threshold queries per agent-object pair, minimizing communication overhead.
• Construct matching algorithms that prioritize matchings satisfying a target economic efficiency property (e.g., Pareto optimality, rank-maximality) while maximizing social welfare.
• Apply theoretical analysis to bound the approximation ratio of the resulting matching relative to the optimal welfare among all matchings satisfying the efficiency constraint.
• Use utility function constructions and worst-case instance design to prove lower bounds on the approximation ratio, showing that Ω(√n) is the best possible for non-adaptive algorithms.

Experimental results

Research questions

• RQ1 Can a small number of threshold queries significantly improve welfare in one-sided matching compared to purely ordinal mechanisms?
• RQ2 What is the best possible approximation ratio to optimal welfare that can be achieved using non-adaptive threshold queries while preserving a given economic efficiency property?
• RQ3 How does the performance of adaptive versus non-adaptive query strategies compare in terms of welfare approximation and query efficiency?
• RQ4 Can the use of threshold queries lead to a constant-factor welfare approximation even when only one bit of cardinal information is collected per agent-object pair?
• RQ5 What are the fundamental limits of welfare approximation in one-sided matching when restricted to simple threshold queries?

Key findings

• The paper establishes that any non-adaptive algorithm using at most one threshold query per agent-object pair cannot achieve a welfare approximation ratio better than Ω(√n) relative to the optimal matching satisfying a given efficiency property.
• For adaptive algorithms, the paper shows that a welfare approximation factor of O(√n) is achievable, matching the lower bound, indicating near-optimality.
• Even with just one bit of cardinal information per object, the proposed algorithms can significantly improve social welfare compared to purely ordinal matching mechanisms.
• The results hold across multiple notions of economic efficiency, including Pareto optimality, rank-maximality, and max-cardinality rank-maximality.
• The theoretical lower bound of Ω(√n) for non-adaptive algorithms is proven via a carefully constructed worst-case instance with specific utility functions and threshold values.
• The framework is extensible to settings with object capacities, requiring only minor modifications such as duplicating object nodes in the matching graph.