Explorations to refine Aizerman Malishevski's representation for path independent choice rules

Abstract

This dissertation consists of four main parts in which we explore Aizerman Malishevski's representation result for path independent choice rules. Each path independent choice rule is known to have a maximizer-collecting (MC) representation: There exists a set of priority orderings such that the choice from each choice set is the union of the priority orderings' maximizers (Aizerman and Malishevski, 1981). In the first part, we introduce the maximal and prime sets to characterize all possible MC representations and show that the size of the largest anti-chain of primes determines its smallest size MC representation. In the second part, we focus on q-acceptant and path independent choice rules. We introduce prime atoms and prove that the number of prime atoms determines the smallest size MC representation. We show that q-responsive choice rules require the maximal number of priority orderings in their smallest size MC representations among all q-acceptant and path independent choice rules. In the third part, we aim to generalize q-responsive choice rules and introduce responsiveness as a choice axiom. In order to provide a new representation for responsive and path independent choice rules we introduce weighted responsive choice rules. Then, we show that all responsive and path independent choice rules are weighted responsive choice rules with an additional regularity condition. In the final part we focus on assignment problem. In this problem Probabilistic Serial assignment is always sd-efficient and sd-envy-free. We provide a sufficient and almost necessary condition for uniqueness of sd-efficient and sd-envy-free assignment via a connectedness condition over preference profile.