Identifying and Quantifying (Un)Improvable Students

2024-07-29

AI Summary

The Deferred Acceptance (DA) mechanism for student matching suffers from Pareto inefficiency. Method: This paper formally defines and quantifies “non-improvable students”—those who cannot achieve a strictly better match under any Pareto-improving mechanism—by constructing a DA-induced envy digraph and analyzing its strongly connected components. Contribution/Results: We prove the digraph contains a unique giant strongly connected component, implying that asymptotically almost all students are improvable. Moreover, we establish the existence of a collection of pairwise-disjoint cycles enabling global, synchronous Pareto improvement. Integrating tools from game theory, graph theory, and matching theory, our analysis uncovers the structural origin of DA’s inefficiency and constructs a mechanism-design foundation for large-scale coordinated optimization. This work provides a novel paradigm for fair and efficient school admissions matching.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Security and Privacy: Large-scale security measurements

Abstract

The Deferred Acceptance (DA) mechanism can generate inefficient placements. Although Pareto-dominant mechanisms exist, it remains unclear which and how many students could improve. We characterize the set of unimprovable students and show that it includes those unassigned or matched with their least preferred schools. Nevertheless, by proving that DA's envy digraph contains a unique giant strongly connected component, we establish that almost all students are improvable, and furthermore, they can benefit simultaneously via disjoint trading cycles. Our findings reveal the pervasiveness of DA's inefficiency and the remarkable effectiveness of Pareto-dominant mechanisms in addressing it, regardless of the specific mechanism chosen.

Problem

• Identifying which students can improve under Deferred Acceptance mechanism
• Quantifying the number of improvable students in school matching
• Analyzing efficiency of Pareto-dominant mechanisms for student placement

Innovation

• Characterizing unimprovable students in DA mechanism
• Proving DA's envy digraph contains giant component
• Establishing Pareto-dominant mechanisms via trading cycles