UniversityAdmissions “Shortlist Matching”

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

2. Matching TheoryGale 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

3. 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

4. “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

5. Turkey Students :1 800 000 take Exam 800 000 qualify to submit rankings (up to 24 departments) Universities : Exam Score-Type (80%) + GPA (20%) 5 Exam Score-Types Quota Total = 200 000 + 200 000 + 200 000 Placement : Gale-Shapley U-Optimal Algorithm Full Scope, Binding China : 35 000 000 Placement : “School Choice” Algorithm : Priority to Students whoTop Rank

6. 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

7. 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 allothers or somehigher (lower) ranked or no corruption control & only suggestive

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

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

10. 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

11. Proposition : Pure Strategy Nash Equilibrium holds in very special cases.

12. No Pure Strategy Equilibrium • If m3 does not interview, thenm3gets w3with probability ¾, it isbetter for m4to interview(w4,w5). • If m4 interviews (w4,w5), it is better for m3to interview(w3,w4). (because m4 will toplist w3 so m3 can get w4+ when w3-.) • If m3 interviews (w3,w4), it is better for m4to interview (w3,w4). (because then with prob ¼,m4 gets w4+ when w3- but if m4 interviews (w4,w5), with prob 3/16 he will get w5+ when w4-.) • If m4 interviews (w3,w4), it is better for m3 not to interview.

13. No Pure Strategy Nash Equilibrium : Idiosyncratic CaseEm4(w4w5|m3(w4w5)) = α + (1- α) p(1-p) ≥ (1- α) (1-p) = Em4(w5w6|m3(w4w5))Em3(w3w4|m4(w3w4)) = α≥ (1- α) (1-p(1-p)) = Em3(w4w5|m3(w3w4))Em4(w5w6|m3(w3w4)) = (1-α) + α p(1-p) ≥ α(1-p(1-p)) = Em4(w4w5|m3(w3w4)) Em3(w3w4|m4(w4w5)) = (1-α) ≥ α =Em3(w2w3|m4(w4w5)) p=1/2, α =7/16

14. shortlisted but unmatched One can continue and match the unmatched with Question : Likelihood of being matchedwithin ? Benchmark:

15. Proposition:

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

18. worst case n=15 k=3

19. circular

20. circular

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

27. worst case n=12 k=3

29. circular

30. circular

31. k = 3 • n almost decomposing circular worst case • 9 8 8 • 60 45 40 36

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