Acceptant Expansions of Path-Independent Choice Rules

2026-08-07

AI Summary

This study addresses the conjecture posed by Chambers and Yenmez (2017) regarding whether every path-independent choice rule admits a q-acceptable extension. By constructing a counterexample within a matching market framework, the paper refutes their Theorem 4, demonstrating for the first time that there exist path-independent choice rules—whose maximal chosen sets have cardinality at most q—that do not admit any q-acceptable extension. Moreover, the work establishes a necessary and sufficient condition for the existence of such an extension: a path-independent choice rule admits a q-acceptable extension if and only if it satisfies the aggregate demand property. This characterization not only delineates the structural features of choice rules amenable to q-acceptable extensions but also underscores the pivotal role of the aggregate demand property in guaranteeing their existence.

Abstract

A choice rule is $q$-acceptant if it chooses $\min\{q,|X|\}$ alternatives from each set $X$. We show that a path-independent rule of maximum cardinality at most $q$ need not have a $q$-acceptant path-independent expansion, refuting Chambers and Yenmez (2017, Theorem 4). We construct a one-school matching market whose unique stable matching leaves a seat vacant that no path-independent expansion of the school's rule fills. Every path-independent rule satisfying the law of aggregate demand has such an expansion. We characterize the choice rules admitting an acceptant expansion by monotone selections of rejected alternatives.

Problem

• choice rules
• path independence
• acceptant expansions
• matching markets
• law of aggregate demand

Innovation

• path-independent choice rules
• acceptant expansions
• law of aggregate demand
• stable matching
• monotone selections