Thesis advisor: Utku Unver / Thesis advisor: Bumin Yenmez / This dissertation consists of two chapters. Both are centered around the theory and design of markets, in which the use of money is prohibited and/or strongly undesirable. In my first chapter, I study multi-object assignment problems. Here, the assignment of graduate students to teaching assistant positions over the course of two semesters, serves as an illustrative application. In my second chapter, I propose an alternative way to distribute asylum seekers among European member states based on the preferences of both sides. Chapter 1: Multi-Object Assignment: Booster Draft In my first chapter, I ask the question of how to divide among a set of n individuals a set of n × m indivisible objects without using monetary transfers, in a way that is efficient, incentive compatible, and ex-post fair. A well known impossibility result shows that the only mechanisms that are both incentive compatible and efficient are dictatorship mechanisms. I fill a gap in the literature by describing a novel mechanism that is both incentive compatible and fair in the responsive preference domain. The mechanism is inspired by booster drafts used in competitive card game tournaments. The idea is to arbitrarily divide the set n × m objects into m \boosters" (sets) of size n and specify a priority order for each such booster. Afterwards the individuals pick objects from the boosters in order of priority. The outcome of the booster draft mechanism can be improved if additional knowledge about a particular market is incorporated into the creation of boosters. I point out a special case of multi-object assignment problems, motivated by the allocation of teaching assignments among graduate students. In this domain the creation of the boosters is straightforward. Indeed, at the Boston College economics department, graduate students are assigned exactly one fall and one spring semester task over the academic year. Here the optimal way of creating boosters is to group up all spring teaching assignments in one booster and all fall semester assignments in the other. In this case the balanced booster draft is not only strategyproof and fair, but also weakly efficient (dominance efficient). Moreover, for this restricted assignment domain I characterize the set of all booster drafts as any (strongly) strategyproof, neutral and non-bossy mechanism. In the final part of the paper I take a closer look at the teaching assistant assignment problem, using date on the submitted rankings over semester-tasks by graduate students. The simulation exercise provides additional evidence that the proposed mechanism is a sensible practical solution. In particular, I show that for a simple measure of welfare students prefer a balanced booster draft to a serial dictatorship mechanism if they are mildly risk averse. Chapter 2: An Alternative Asylum Assignment The 2015 refugee crisis has demonstrated the necessity of revising the current European asylum system. As an alternative, I propose to take into account preferences of asylum seekers as well as preferences of member states. Asylum seekers indicate how long they are willing to wait for their asylum application for any given member state, allowing them to avoid overburdened member states by opting for \less popular" member states. Within the market design literature, this is the first paper proposing to match asylum seekers as opposed to refugees. In other words, its stays much closer to the template of the Common European Asylum System. From a theoretical perspective, it turns out that the asylum seeker framework can be formulated as an application of the well-known matching with contracts model by Hatfield and Milgrom (2005a). This simplifies the analysis a great deal, as matching with contracts is a well studied framework within the matching/market design literature. I show that the standard cumulative offer mechanism (Gale and Shapley, 1962a; Hatfield and Kojima, 2010a) is asylum seeker incentive compatible and leads to stable outcomes, using the fact that the proposed choice functions have a completion satisfying substitutability and the law of aggregate demand Hatfield and Kominers (2016). Moreover, stability implies two sided Pareto efficiency, giving consideration to both preferences of member states and asylum seekers. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.