On Acceptant and Substitutable Choice Rules ∗ Battal Do˘ gan † Serhat Do˘ gan ‡ Kemal Yıldız § September 10, 2018 Abstract Each acceptant and substitutable choice rule is known to have a maximizer-collecting represen- tation: 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 representa- tion. 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 charac- terize 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. JEL Classification Numbers: D01, D03, C78, D47, D78. Keywords: Choice rules, acceptance, substitutability, path independence, prime atom. ∗ Battal Do˘ gan gratefully acknowledges financial support from the Swiss National Science Foun- dation (SNSF) and Kemal Yıldız gratefully acknowledges financial support from the Scientific and Research Council of Turkey (TUBITAK). We thank Ahmet Alkan, Kfir Eliaz, Ravi Jagadeesan, Bet- tina Klaus, Scott Duke Kominers, Ilan Nehama, Ariel Rubinstein, Ran Spiegler, William Thomson, anonymous referees, and seminar participants at Harvard University (CMSA), Tel Aviv University, and Advances in Fair Division Conference in St. Petersburg for valuable comments and suggestions. † Department of Economics, University of Bristol; battal.dogan@bristol.ac.uk. ‡ Department of Economics, Bilkent University; dserhat@bilkent.edu.tr. § Department of Economics, Bilkent University; kemal.yildiz@bilkent.edu.tr. 1