This thesis consists of three papers on market design which address broadly applicable questions on the design of school choice mechanisms, refugee placement, assignment in entry-level labour markets and similar matching rules.In the  first paper a new family of rules is introduced for many-to-one matching problems, the Preference Rank Partitioned (PRP) rules. PRP rules are Student-Proposing Deferred Acceptance rules where the schools use a choice function based on the students' preference orderings in addition to the schools' strict priority orderings. Each PRP rule uses a choice function which is a function of a fixed partition of both student preference ranks and school priority ranks: the choice function first seeks to select students based on the priority classes and then based on the preference classes. The strict priorities are only used for tie-breaking. PRP rules include many well-known matching rules and some interesting new rules, and we analyze them in this unified framework.
In the second paper we study a new class of matching rules, called Deferred Acceptance with Improvement Trading Cycles (DA-ITC), which start with the DA, and if the DA outcome is not Pareto-efficient then there is an iterated improvement trading cycle phase which allows for Pareto-improvements until a Pareto-efficient outcome is reached. We first revisit EADAM (Kesten, 2010) and show that a simple algorithm which retraces cycles in the DA procedure in a backward order of the rejections is equivalent to the EADAM rule. The new class of DA-ITC rules contains the EADAM and DA-TTC as its two extreme members and exhibits some of their desirable properties.
In the third paper we focus on matching problems where stability need not be satisfied if the violation of priorities is "small," such as when a small priority difference is considered insignificant or when one is willing to consent but only if the priority reversal is small. Based on the degree of stability which specifies what is considered a small priority gap, we define two families of matching rules, the k-Consent rules and the k-DA rules, and explore their attributes.