Acceptant Expansions of Path-Independent Choice Rules

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.