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UniversityAdmissions “Shortlist Matching” Challenges in the Light of Matching Theory and Current Practises 24 May 2011 HSE Ahmet Alkan Sabancı University

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Matching Theory Gale Shapley, ‘College Admissions and Stability of Marriage’ American Mathematical Monthly, 1962 A matching is an allocation of students to universities. stable if there is no student s who would rather be at university U and university U would rather replace one of its students or an empty slot with s. Solution Concept, Benchmark Model : most successful

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Institutions Decentralized “university admissions” Market U.S.A Centralized Marketplace Institution “Student Selection Assesment and Placement” Turkey, China, … Semi-Centralized Marketplace Institution “National Intern Resident Matching Program” U.S.A Two-Stage Decentralized ‘Shortlist’ + Centralized Final Matching

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“National Intern Resident Matching Program” and similar marketplace institutions studied intensively Alvin Roth and collaborators Hailed over decentralized markets for mainly 2 efficiency attributes: All Together (Scope) All at Same Time (Coordination) Inefficiencies in decentralized matching : bounded search congestion unravelling

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Turkey Students : take Exam qualify to submit rankings (up to 24 departments) Universities : Exam Score-Type (80%) + GPA (20%) 5 Exam Score-Types Quota Total = Placement : Gale-Shapley U-Optimal Algorithm Full Scope, Binding China : Placement : “School Choice” Algorithm : Priority to Students whoTop Rank

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Turkey EXAM woes : Incentives on Pre-University Education Poor / Narrow “You get what you measure” ‘Classroom’ Drilling Sector twice the budget of all universities Equity How to restore quality in Pre-University Schools

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Centralized Two-Stage Matching Shortlist + Final Incentives on Pre-university Education : restore domain where middle and high schools can perform and compete for excellence Avoid Inefficiencies inherent in Decentralization save further on search Control for Corruption restrict match to shortlist or shortlist plus all others or some higher (lower) ranked or no corruption control & only suggestive

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~ exam score gpa age location endowment depth maturity drive warmth beauty ~ Model

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Centralized Two-Stage Matching Shortlist + Final Find many-to-many matching on shortlist of Invite to submit Find stable matching on

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Objections : not constitutional all the extra work for University Admissions Offices corruption but : “why not decentralize completely as in the US” how to shortlist : instability ? shortlisted but unmatched

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Proposition : Pure Strategy Nash Equilibrium holds in very special cases

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No Pure Strategy Equilibrium If m 3 does not interview, then m 3 gets w 3 with probability ¾, it is better for m 4 to interview (w 4,w 5 ). If m 4 interviews (w 4,w 5 ), it is better for m 3 to interview (w 3,w 4 ). (because m 4 will toplist w 3 so m 3 can get w 4+ when w 3-. ) If m 3 interviews (w 3,w 4 ), it is better for m 4 to interview (w 3,w 4 ). (because then with prob ¼, m 4 gets w 4+ when w 3- but if m 4 interviews (w 4,w 5 ), with prob 3/16 he will get w 5+ when w 4-.) If m 4 interviews (w 3,w 4 ), it is better for m 3 not to interview. w1w1 w2w2 w3w3 w4w4 w5w5 m1m1 11 m2m2 2 m3m m4m4 211

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No Pure Strategy Nash Equilibrium : Idiosyncratic Case Em 4 (w 4 w 5 |m 3 (w 4 w 5 )) = α + (1- α) p(1-p) ≥ (1- α) (1-p) = Em 4 (w 5 w 6 |m 3 (w 4 w 5 )) Em 3 (w 3 w 4 |m 4 (w 3 w 4 )) = α ≥ (1- α) (1-p(1-p)) = Em 3 (w 4 w 5 |m 3 (w 3 w 4 )) Em 4 (w 5 w 6 |m 3 (w 3 w 4 )) = (1-α) + α p(1-p) ≥ α (1-p(1-p)) = Em 4 (w 4 w 5 |m 3 (w 3 w 4 )) Em 3 (w 3 w 4 |m 4 (w 4 w 5 )) = (1-α) ≥ α = Em 3 (w 2 w 3 |m 4 (w 4 w 5 )) p=1/2, α =7/16 w1w2w3w4w5w6 m1 11 m2 221 m3 212 m4 211

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shortlisted but unmatched Benchmark: One can continue and match the unmatched with Question : Likelihood of being matched within ?

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Proposition:

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minimum maximal matching for k-regular bipartite graph B(n,k) sharp Yannakakis and Gavril 1978 NP-hard even when max degree is 3 cardinality

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worst case n=15 k=3

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circular

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Proposition The minimum maximal matching cardinality for circular B(n,3) is 2/3 n for n multiple of 3(k-1).

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worst case n=12 k=3

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circular

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n almost decomposing circular worst case k = 3

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Concluding Remarks Two-Stage Mechanism to improve efficiency scope coordination information acquisition incentives for pre-university education with levers to control for corruption.

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