Unraveling in Matching Markets with Distributional Constraint Interaction of Early Admission and Centralized College Admission in China Yuqing Hu University of Southern California, yuqingh@usc.edu
Research Questions ´ What causes unraveling? Why do some matching markets suffer from unraveling? But some do not? ´ How do we prevent unraveling in matching markets? ´ Early contract in labor markets ´ Early admission in school choice ´ … ´ In school choice, how does decentralized admission interact with centralized admission? ´ In many-to-one matching markets with distributional constraint (max quota in China), how does unraveling show different patterns?
Main Results ´ We propose a new mechanism that we call ´ “Trading cycle with deferred acceptance” (TCDA) ´ Or “bi-deferred acceptance” (BDA) ´ Mimic the decentralized matching process ´ Reasons for different unraveling equilibria ´ Unstable unraveling equilibrium due to competition ´ Preference similarity ´ Easy to overcome ´ Stable unraveling equilibrium due to inefficiency ´ Cyclic preference is the source of inefficiency for the proposed side ´ How do we measure inefficiency? the last “large” cycle in TCDA/BDA
Main Results ´ Reasons for different matching markets to unravel ´ Blocking pairs in an unstable matching market want to go early ´ Partial unraveling ´ The proposed side in a stable matching market want to go early ´ Full unraveling ´ Acyclic preference: no blocking pairs, no inefficacy ´ TCDA can reduce full unraveling
Background: 1000 Mainland China’s college 800 Number in 10000s admission 600 ´ Very competitive 400 200 0 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 2016 year Applied Admitted Figure 1. College Admission from 1977 to 2015
Background: 1,000 Mainland China’s college 800 Number in 10000s admission 600 ´ Very competitive 400 200 0 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Total Applied Admitted Admitted to 4-year colleges Admitted to 1st tier colleges Figure 2. College Admission from 2006 to 2015
Background: Mainland China’s Heilongjiang college Jilin Inner Mongol admission Xinjiang Beijing Liaoning Liaoning Liaoning Liaoning Liaoning Hebei Hebei Tianjin ´ Very competitive Gansu Shanxi Shandong Shandong Ningxia ´ Large regional inequality Qinghai Shaanxi Henan Tibet Jiangsu Anhui Shanghai Shanghai Shanghai Hubei Sichuan Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Chongqing Jiangxi Hunan Guizhou Fujian Fujian Fujian Fujian Fujian Fujian Fujian Yunnan Guangxi (51.88,87.88] Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong (34.49,51.88] (23.9,34.49] [1.5,23.9] No data Hainan Figure 3. Number of Applicants in 2007 (in 0,000s)
Background: Mainland China’s Heilongjiang college Jilin Inner Mongol admission Xinjiang Beijing Liaoning Liaoning Liaoning Liaoning Liaoning Hebei Hebei ´ Very competitive Tianjin Gansu Shanxi Shandong Shandong Ningxia ´ Large regional inequality Qinghai Shaanxi Henan Tibet Jiangsu Anhui Shanghai Shanghai Shanghai Hubei Sichuan Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Chongqing Jiangxi Hunan Guizhou Fujian Fujian Fujian Fujian Fujian Fujian Fujian Yunnan Guangxi (8.24,60.61] Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong (6.64,8.24] (4.89,6.64] [.19,4.89] No data Hainan Figure 4. Admission Rate in 2007
Background: Mainland China’s college admission ´ Very competitive ´ Large regional inequality ´ Two admission channels ´ Centralized ´ Reforms: from sequential mechanism (2003) to parallel mechanism (2015) to DA ´ Decentralized ´ Key universities (about 100), special schools (arts, military school)
Background: Mainland China’s college admission ´ Centralized college admission reform Sequential mechanism before 2003 until 2015 Parallel mechanism massive reform in started in 2003 2008 Gale-Sharply DA algorithm Figure 5. Application Rate in 2007 (in 0,000s) massive reform in started in 2015 2017
Background: Number of Colleges With Decentralized Admission Mainland China’s 100 college admission sequential massive parallel deferred mechanism mechanism reform acceptance (paraellel in 2008 and 2009 started in 2015 mechanism massive DA 80 started in reform in 2016 ´ Decentralized college one province and 2017 number of colleges admission in 2003) 60 Admission before NCEE early admission college collusion no more collusion started in 2003 until 2015 started started in 2009 40 no more early admission College collusion 20 started in 2009 until 2015 0 2002 2004 2006 2008 2010 2012 2014 2016 2018 Admission after NCEE year started in 2015 Figure 6. Number of Colleges Participating in Decentralized Admission
Background: Mainland China’s colleges National college admission propose to college en- students take Students students trance exam colleges’ indi- submit prefer- (Baosong) (Gaokao) vidual exams ences ´ Decentralized college admission Admission before NCEE t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 started in 2003 until 2015 College collusion started in 2009 until 2015 students ap- early admis- Gaokao re- Centralized ply to individ- sion sults come matching ual colleges out Admission after NCEE started in 2015 Figure 7. Admission before 2015
Background: Mainland China’s students ap- students take Gaokao re- college admission ply to individ- Centralized colleges’ indi- sults come ual colleges matching vidual exams out ´ Decentralized college admission Admission before NCEE t 1 t 2 t 3 t 4 t 5 t 6 t 7 started in 2003 until 2015 College collusion National conditional Students started in 2009 until 2015 college en- admission submit prefer- trance exam ences Admission after NCEE (Gaokao) started in 2015 Figure 8. Admission after 2015
Why does unraveling occur in a matching market? ´ Consider different mechanisms: ´ Top-trading cycle ´ Serial dictatorship ´ Immediate-acceptance algorithm (Boston mechanism) ´ Deferred-acceptance algorithm ´ Trade-off between efficiency, stability and strategy- proofness
Why does unraveling occur in a matching market? ´ Trading cycle with deferred acceptance, or Bi-deferred acceptance ´ Round 1: ´ M è W, W è M, let permanent matching occur when m çè w, and remove those pairs ´ M or W with multiple proposals: only keep the best one and reject the rest ´ Round n: ´ Rejected M and W propose to their next favorite ones ´ M è W, W è M, let permanent matching occur when m çè w, and remove those pairs ´ M or W with multiple proposals: only keep the best one and reject the rest ´ Iteration stops when there are no more rejected agents on at least one side ´ For the remaining cycles, let the W(M) in the cycle choose their favorite M(W); or randomly break the ties.
Why does unraveling occur in a matching market? ´ A simple example of TCDA in a one-to-one market ´ W= { w 1 , w 2 , w 3 , w 4 } , M= { m 1 , m 2 , m 3 , m 4 } ; ´ m 1 : w 1 ≻ w 3 ≻ w 2 ≻ w 4 ; w 1 : m 2 ≻ m 4 ≻ m 3 ≻ m 1 ; ´ m 2 : w 3 ≻ w 2 ≻ w 1 ≻ w 4 ; w 2 : m 1 ≻ m 3 ≻ m 2 ≻ m 4 ; ´ m 3 : w 1 ≻ w 4 ≻ w 3 ≻ w 2 ; w 3 : m 3 ≻ m 1 ≻ m 2 ≻ m 4 ; ´ m 4 : w 4 ≻ w 2 ≻ w 1 ≻ w 3 ; w 4 : m 2 ≻ m 3 ≻ m 4 ≻ m 1 .
Why does unraveling occur in a matching market? ´ W= { w 1 , w 2 , w 3 , w 4 } , M= { m 1 , m 2 , m 3 , m 4 } ; ´ m 1 : w 1 ≻ w 3 ≻ w 2 ≻ w 4 ; w 1 : m 2 ≻ m 4 ≻ m 3 ≻ m 1 ; ´ m 2 : w 3 ≻ w 2 ≻ w 1 ≻ w 4 ; w 2 : m 1 ≻ m 3 ≻ m 2 ≻ m 4 ; ´ m 3 : w 1 ≻ w 4 ≻ w 3 ≻ w 2 ; w 3 : m 3 ≻ m 1 ≻ m 2 ≻ m 4 ; ´ m 4 : w 4 ≻ w 2 ≻ w 1 ≻ w 3 ; w 4 : m 2 ≻ m 3 ≻ m 4 ≻ m 1 . ´ Round 1 : m 1 è w 1 , m 2 è w 3 , m 3 è w 1 , m 4 è w 4 ; w 1 è m 2 , w 2 è m 1 , w 3 è m 3 , w 4 è m 2 .
Why does unraveling occur in a matching market? ´ W= { w 1 , w 2 , w 3 , w 4 } , M= { m 1 , m 2 , m 3 , m 4 } ; ´ m 1 : w 1 ≻ w 3 ≻ w 2 ≻ w 4 ; w 1 : m 2 ≻ m 4 ≻ m 3 ≻ m 1 ; ´ m 2 : w 3 ≻ w 2 ≻ w 1 ≻ w 4 ; w 2 : m 1 ≻ m 3 ≻ m 2 ≻ m 4 ; ´ m 3 : w 1 ≻ w 4 ≻ w 3 ≻ w 2 ; w 3 : m 3 ≻ m 1 ≻ m 2 ≻ m 4 ; ´ m 4 : w 4 ≻ w 2 ≻ w 1 ≻ w 3 ; w 4 : m 2 ≻ m 3 ≻ m 4 ≻ m 1 . ´ Round 1 : m 1 è w 1 , m 2 è w 3 , m 3 è w 1 , m 4 è w 4 ; w 1 è m 2 , w 2 è m 1 , w 3 è m 3 , w 4 è m 2 .
Why does unraveling occur in a matching market? ´ W= { w 1 , w 2 , w 3 , w 4 } , M= { m 1 , m 2 , m 3 , m 4 } ; ´ m 1 : w 1 ≻ w 3 ≻ w 2 ≻ w 4 ; w 1 : m 2 ≻ m 4 ≻ m 3 ≻ m 1 ; ´ m 2 : w 3 ≻ w 2 ≻ w 1 ≻ w 4 ; w 2 : m 1 ≻ m 3 ≻ m 2 ≻ m 4 ; ´ m 3 : w 1 ≻ w 4 ≻ w 3 ≻ w 2 ; w 3 : m 3 ≻ m 1 ≻ m 2 ≻ m 4 ; ´ m 4 : w 4 ≻ w 2 ≻ w 1 ≻ w 3 ; w 4 : m 2 ≻ m 3 ≻ m 4 ≻ m 1 . ´ Round 1 : m 1 è w 1 , m 2 è w 3 , m 3 è w 1 , m 4 è w 4 ; w 1 è m 2 , w 2 è m 1 , w 3 è m 3 , w 4 è m 2 . ´ Chains/Cycles: w 3 è m 3 è w 1 è m 2 è w 3 ; m 4 è w 4 ; w 2 è m 1 .
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