Two-Body Assignment Problem in the Context of the Israeli Medical Internship Match

This thesis was submitted as partial fulfillment of the requirements for the Masters Degree in the Department of Computer Science, Bar-Ilan University. The final step in getting an Israeli M.D. is performing a year-long internship in one of the hospitals in Israel. Internships are decided upon by a lottery, which is known as "The Internship Lottery". In 2014 we redesigned the lottery, replacing it with a more efficient one. The new method is based on calculating a tentative lottery, in which each student has some probability of getting to each hospital. Then a computer program "trades" between the students, where trade is performed only if it is beneficial to both sides. This trade creates surplus, which translates to more students getting one of their top choices. The average student improved his place by $0.91$ seats. The new method can improve the welfare of medical graduates, by giving them more probability to get to one of their top choices. It can be applied in internship markets in other countries as well. This thesis presents the market, the redesign process and the new mechanism which is now in use. There are two main lessons that we have learned from this market. The first is the "Do No Harm" principle, which states that (almost) all participants should prefer the new mechanism to the old one. The second is that new approaches need to be used when dealing with two-body problems in object assignment. We focus on the second lesson, and study two-body problems in the context of the assignment problem. We show that decomposing stochastic assignment matrices to deterministic allocations is NP-hard in the presence of couples, and present a polynomial time algorithm with the optimal worst case guarantee. We also study the performance of our algorithm on real-world and on simulated data.