Three essays on matching theory and application of school choice problems

This thesis consists of three chapters on matching theory and application of school choice problems. The first chapter is a theory essay about kidney exchange literature. We study dynamic mechanisms in multi-hospital kidney exchange. We first consider the combination of two hospitals. We show that for a range of arrival rates of incompatible pairs, hospitals obtain a higher surplus by matching their own pairs internally than participating a combined pool. We design a threshold inter-hospital matching mechanism to match patient-donor pairs in two and three-way exchanges dynamically. This mechanism relies on inter-hospital exchanges only when certain conditions are met. The threshold inter-hospital matching mechanism is preferred to matching its own pairs internally and to participating in a combined pool. In the second chapter, an application of a common enrollment in school choice is analyzed. Centralized admissions for charter, private, and neighborhood schools are commonly used across some countries such as the United States. It is shown theoretically that full participation in the unified enrollment system provides the best outcome for the students when students find all schools acceptable and schools use responsive choice rule (Ekmekçi and Yenmez [2019]). Some schools may prefer to admit students independently since schools have incentives to stay out of the common enrollment system. Thus, some districts have not succeeded in satisfying full participation of their schools to the common enrollment system. It can be analyzed that full participation of schools can be satisfied under some specific timing scenarios for enrollment such as preemption, simultaneous, and after timing scenarios. On the other hand, when the main assumptions are relaxed n a specific simulated environment, the proven theorems (defined in Ekmekçi and Yenmez [2019]) are violated. The real-life school enrollment systems are imitated through using some simulations. It is indicated that not only the preemption timing scenario but also simultaneous timing scenario are integration compatible by putting different correlations on student preferences and school priorities in the simulations. The order of the evading school in the preference list of students is a significant determinant for persuading it to participate in the unified system. The choice between responsive or quota-filling school choice rule is also crucial to forcing the evading school to participate in the common enrollment system. Lastly, the after-timing scenario is not integration compatible for different correlation coefficient values among students under the relaxation of assumptions defined on choice rules in Theorem 4, Ekmekçi and Yenmez [2019]). In the third chapter of this thesis, an application of a theory of student assignment problem, inter-district school choice where a student can be allotted to a school outside of her district is analyzed. The model is a matching with contracts between students and districts where a contract determine the specific school within the district that the student applies. Four policy goals are taken into account: individual rationality, improving student welfare, balanced exchange policy and student diversity in each district. The balanced exchange policy is that the number of students that each district admits from the other district must be equal to the number of students that it transfers to the others. We simulate the real-life scenarios by embedding different positive coefficients into the utility functions of each student defined for each school and vice versa. Then we use these utilities to construct preference list of students and priority ordering of schools. Since, there is a tradeoff between satisfying the stability of the matching and balancing the student diversity in each district, this study measures the effect of diversification among district on the welfare of different type of students under balanced exchange policy. It is observed whether the theorems proven in Hafalir et al. [2022] still hold or not under the relaxation of the assumptions or putting some constraints.