The Mathematics and Economics of Matchings Emma Kulek, Sebastian Cioaba Department of Mathematical Sciences, University of Delaware, Newark, DE 19716

Abstract Initial Matching Procedures Conclusion The 2012 Nobel Prize in Economic Sciences awarded to Alvin Roth and Lloyd Shapley inspires this Both the New York City and Boston Public School matching research project. As one of the irst to study cooperative game theory, Shapley aimed to determine how NYC individuals might cooperatively choose an allocation. In such, the idea of stability is a central concept. Shapley u Over 500 programs (Specialized, Unscreened, Zoned, Screened, EdOpt) mechanisms needed to be changed. Each school system had and his colleague devised the Gale-Shapley algorithm, which solves the Stable Marriage Problem and always results in a stable matching. The concept of the Gale-Shapley algorithm has been applied to many important u 100,000 students, 5 programs max, preference list different characteristics and therefore needed to be studied real-world situations. It has aided the matching process of medical residents to hospitals, college applicants to u Schools use ranking as admission criteria colleges, compatible kidney donors to recipients, New York City public school students to high schools, Boston separately. The NYC public school system is a two-sided public school students to high schools, and many more. The implementation of a new matching system was u Applicant receives letter with decision (accepted, waitlisted, rejected) crucial for the NYC and Boston public school systems. To implement such a system successfully, it was u Accepts no more than one offer and one waitlist market with both students and schools possessing important to understand what was wrong with the, then current, system. Both public school systems appeared u Repeated- second and third letter preferences and acting as strategic players. In the Boston to be two-sided markets in which the schools, and the students had varying preferences. The consequential u Congestion dificulty was inding a stable assignment such that no school and student were not matched to one another, public school system, schools preferences are set centrally, who both preferred to be. The resulting success suggests that the employment of such a matching system could u About 50,000 got offers initially (17,000- multiple) beneit a wide range of markets. u 30,000 assigned to schools not on preference list making students the only strategic players. In the end, u Risk associated with ranking school irst (if possibility of rejection) different variations of the Gale-Shapley deferred acceptance u Schools were strategic- concealed capacity algorithm changed the way students and schools were Background Boston matched together. Theorem Let G be any bipartite graph with ranking matrices L and u 60,000 students k-12, 140 schools, 3 zones (East, West, North) u First registration- students entering new school rank at least 3 (order of preference) R. Then G has a stable matching which is a maximal matching in G. u System of priorities u Younger sibling- priority to same schools as older A Bipartite Graph is a graph whose vertices u Walk zone (half of seats, only irst registration) Future Work u Neighborhoods without walk zone- priority to 1st and 2nd choice can be divided into two disjoint sets u Additional priorities- random numbers generated one for each student Although the Gale-Shapley algorithm has assisted the u Priority Matching Mechanism aiming to match students to 1st choice advancement of many matching mechanisms, there are a lot st A Matching is a collection of disjoint u Step 1– School considers students who listed it as 1 choice, assign seats in of additional ways in which it could be applied: priority order edges • Assisting the passage of data packets through a series of u Step k– Each school (with seats available) considers students who listed it as kth choice, assign seats in priority order A Maximal Matching is a matching that routers from one computer to another, controlling internet u Risk associated with ranking schools irst cannot be enlarged by adding an edge trafic • Determining which lights get priority landing during A Perfect Matching is a matching which inclement weather when an airline cancels a light contains every vertex of the graph Solutions & Results • Understanding and predicting the success of online dating

Goal: to create strategy-proof mechanisms to match students to schools A Cycle is a closed path in which the irst

and last vertices are the same The New York City High School Match (2003) The Boston Public School Match (2006) References

Deferred Acceptance Algorithm Two alternative strategy-proof mechanisms: 1. West, Douglas Brent. Introduction to Graph Theory. Upper 1. Deferred Acceptance Saddle River, NJ: Prentice Hall, 2001. Print. Gale-Shapley Algorithm 2. Top Trading Cycles 2. Abdulkadiroğlu, Atila, Parag A. Pathak, and Alvin E. Roth. "The New York City High School Match." American The Gale-Shapley algorithm (David Gale, Lloyd Shapley 1962) Economic Review 95.2 (2005): 364-67. Web. inds a solution to the stable marriage problem. The 3. Abdulkadiroğlu, Atila, Parag A. Pathak, Alvin E. Roth, and algorithm includes a number of iterations, which eventually o Two-sided market Tayfun Sönmez. "The Boston Public School Match." o Preference list- 12 schools max American Economic Review 95.2 (2005): 368-71. Web. lead to a stable match (marriage.) o No multiple offers o Student proposing- state true preferences A B C o Unassigned after round 2- new list of o Importance of truth telling schools o Preference list- unlimited number of α 1,3 2,2 3,1 schools Acknowledgements ² Over 70,000 matched to school on initial Priorities ixed in advance: one agent β 3,1 1,3 2,2 o preference list making choices γ 2,2 3,1 1,3 Summer Scholars Program ² Only 3,000 did not receive any school they o July 2005- voted for deferred- Ranking matrix of three men, α, β, and γ, and three women, A, B, and C. chose acceptance Ofice of Undergraduate Research and Experiential Learning