The extended serial correspondence on a rich preference domain

We study the problem of assigning objects to a set of agents. We focus on probabilistic solutions that only take agents' preferences over objects as input. Importantly, agents may be indifferent among several objects. The "extended serial correspondence" is proposed by Katta and Sethuraman (J Econ Theory 131:231-250, 2006) to solve this problem. As a follow-up to Liu and Pycia (Ordinal efficiency, fairness, and incentives in large markets. Mimeo, 2012) who introduce the notion of profiles with "full support", we work with two interesting classes of preference profiles: profiles that (i) have rich support on a partition or (ii) are single-peaked with rich support on a partition. For each profile in these classes, an assignment matrix is selected by the extended serial correspondence if and only if it is sd-efficient and sd envy-free. We also provide an asymptotic result.