On Acceptant and Substitutable Choice Rules
34 Pages Posted: 20 Jul 2017 Last revised: 23 Sep 2018
Date Written: September 10, 2018
Each acceptant and substitutable choice rule is known to have a maximizer-collecting representation: there exists a list of priority orderings such that from each choice set that includes more elements than the capacity, the choice is the union of the priority orderings' maximizers (Aizerman and Malishevski, 1981). We introduce the notion of a prime atom and constructively prove that the number of prime atoms of a choice rule determines its smallest size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their maximizer-collecting representations among all acceptant and substitutable choice rules. We characterize maximizer-collecting choice rules in which the number of priorities equals the capacity. We also show that if the capacity is greater than three and the number of elements exceeds the capacity by at least two, then no acceptant and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.
Keywords: Choice rules, acceptance, substitutability, path independence, prime atom
JEL Classification: D01, D03, C78, D47, D78
Suggested Citation: Suggested Citation