Unraveling in Matching Markets with Distributional Constraint - - PowerPoint PPT Presentation

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

200 400 600 800 1000 Number in 10000s 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 2010 2013 2016 year Applied Admitted

Background: Mainland China’s college admission

´ Very competitive Figure 1. College Admission from 1977 to 2015

200 400 600 800 1,000 Number in 10000s 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Total Applied Admitted Admitted to 4-year colleges Admitted to 1st tier colleges

Background: Mainland China’s college admission

´ Very competitive Figure 2. College Admission from 2006 to 2015

Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei Heilongjiang Henan Hubei Hunan Jiangsu Jiangxi Jilin Liaoning Inner Mongol Ningxia Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Tibet Yunnan Zhejiang Fujian Fujian Fujian Fujian Fujian Fujian Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Hebei Liaoning Liaoning Liaoning Liaoning Shandong Shanghai Shanghai Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang

(51.88,87.88] (34.49,51.88] (23.9,34.49] [1.5,23.9] No data

Background: Mainland China’s college admission

´ Very competitive ´ Large regional inequality Figure 3. Number of Applicants in 2007 (in 0,000s)

Background: Mainland China’s college admission

´ Very competitive ´ Large regional inequality Figure 4. Admission Rate in 2007

Anhui Beijing Chongqing Fujian Gansu Guangdong Guangxi Guizhou Hainan Hebei Heilongjiang Henan Hubei Hunan Jiangsu Jiangxi Jilin Liaoning Inner Mongol Ningxia Qinghai Shaanxi Shandong Shanghai Shanxi Sichuan Tianjin Xinjiang Tibet Yunnan Zhejiang Fujian Fujian Fujian Fujian Fujian Fujian Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Guangdong Hebei Liaoning Liaoning Liaoning Liaoning Shandong Shanghai Shanghai Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang Zhejiang

(8.24,60.61] (6.64,8.24] (4.89,6.64] [.19,4.89] No data

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

Gale-Sharply DA algorithm

started in 2015 massive reform in 2017

Parallel mechanism

started in 2003 massive reform in 2008

Sequential mechanism

before 2003 until 2015

Figure 5. Application Rate in 2007 (in 0,000s)

Background: Mainland China’s college admission

´ Decentralized college admission Figure 6. Number of Colleges Participating in Decentralized Admission

Admission after NCEE

started in 2015

College collusion

started in 2009 until 2015

Admission before NCEE

started in 2003 until 2015 sequential mechanism (paraellel mechanism started in

• ne province

in 2003) college collusion started in 2009 no more collusion no more early admission massive parallel mechanism reform in 2008 and 2009 deferred acceptance started in 2015 early admission started massive DA reform in 2016 and 2017 20 40 60 80 100 number of colleges 2002 2004 2006 2008 2010 2012 2014 2016 2018 year

Number of Colleges With Decentralized Admission

Background: Mainland China’s college admission

´ Decentralized college admission Figure 7. Admission before 2015

Admission after NCEE

started in 2015

College collusion

started in 2009 until 2015

Admission before NCEE

started in 2003 until 2015

t1 t2 t3 t4 t5 t6 t7 t8

colleges propose to students (Baosong) students ap- ply to individ- ual colleges students take colleges’ indi- vidual exams early admis- sion National college en- trance exam (Gaokao) Gaokao re- sults come

• ut

Students submit prefer- ences Centralized matching

Background: Mainland China’s college admission

´ Decentralized college admission Figure 8. Admission after 2015

Admission after NCEE

started in 2015

College collusion

started in 2009 until 2015

Admission before NCEE

started in 2003 until 2015

t1 t2 t3 t4 t5 t6 t7

students ap- ply to individ- ual colleges National college en- trance exam (Gaokao) students take colleges’ indi- vidual exams conditional admission Gaokao re- sults come

• ut

Students submit prefer- ences Centralized matching

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:

´ MèW, WèM, let permanent matching occur when mçèw, 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 ´ MèW, WèM, let permanent matching occur when mçèw, 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);

• r randomly break the ties.

Why does unraveling occur in a matching market?

´ A simple example of TCDA in a one-to-one market ´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1.

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 1: m1èw1, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm2.

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 1: m1èw1, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm2.

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 1: m1èw1, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm2. ´ Chains/Cycles: w3èm3èw1èm2èw3; m4èw4; w2è m1.

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 1: m1èw1, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm2. ´ Round 2: m1èw3, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 2: m1èw3, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 2: m1èw3, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 2: m1èw3, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm3 ´ Chains/Cycles: m4è w4è m3è w1è m2; w2è m1è w3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 2: m1èw3, m2èw3, m3èw1, m4èw4; w1èm2, w2èm1, w3èm3, w4èm3 ´ Round 3: m2èw2, w3èm1

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 3: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm1, w3èm1, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 3: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm1, w3èm1, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 3: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm1, w3èm1, w4èm3 ´ Chains/cycles: m1è w3è m1; m4è w4è m3è w1è m2è w2

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 3: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm1, w3èm1, w4èm3 ´ Round 4: w2èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 4: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm3, w3èm1, w4èm3

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 4: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm3, w3èm1, w4èm3 ´ Chains/cycles: m1èw3èm1; m4èw4èm3èw1èm2èw2;

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 4: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm3, w3èm1, w4èm3 ´ Round 5: w2èm2

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 5: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm2, w3èm1, w4èm3 ´ Chains/cycles: m1èw3èm1; m2èw2èm2 ; m4èw4èm3èw1

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 5: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm2, w3èm1, w4èm3 ´ Chains/cycles: m1èw3èm1; m2èw2èm2 ; m4èw4èm3èw1

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 5: m1èw3, m2èw2, m3èw1, m4èw4; w1èm2, w2èm2, w3èm1, w4èm3 ´ Round 6: w1èm4

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ Round 6: m1èw3, m2èw2, m3èw1, m4èw4; w1èm4, w2èm2, w3èm1, w4èm3 ´ Chains/cycles: m1èw3èm1; m2èw2èm2 ; m4èw4èm3èw1èm4 ´ Break the “large” cycle: m4èw4èm3èw1èm4, either m4èw4, m3èw1 or w4èm3, w1èm4

Why does unraveling occur in a matching market?

´ W={w1, w2, w3, w4}, M={m1, m2, m3, m4}; ´ m1: w1 ≻ w3 ≻ w2 ≻ w4; w1: m2 ≻ m4 ≻ m3 ≻ m1; ´ m2: w3 ≻ w2 ≻ w1 ≻ w4; w2: m1 ≻ m3 ≻ m2 ≻ m4; ´ m3: w1 ≻ w4 ≻ w3 ≻ w2; w3: m3 ≻ m1 ≻ m2 ≻ m4; ´ m4: w4 ≻ w2 ≻ w1 ≻ w3; w4: m2 ≻ m3 ≻ m4 ≻ m1. ´ M-optimal TCDA: (m1, w3; m2, w2; m3, w1; m4, w4)

´ W in the last cycle want to manipulate

´ W-optimal TCDA: (m1, w3; m2, w2; m3, w4; m4, w1)

´ M in the last cycle want to manipulate

Why does unraveling occur in a matching market?

• 𝜈""#$ =

&' &) &* &+ ,) ,* ,' ,+

, 𝜈""#. =

&' &) &* &+ ,) ,' ,* ,+

• 𝜈/0$ =

&' &) &* &+ ,) ,* ,' ,+

, 𝜈/0$ =

&' &) &* &+ ,) ,' ,* ,+

• 𝜈10$ =

&' &) &* &+ ,* ,) ,' ,+

, 𝜈10. =

&' &) &* &+ ,* ,) ,+ ,'

• 𝜈"#10$ =

&' &) &* &+ ,* ,) ,' ,+

, 𝜈"#10, =

&' &) &* &+ ,* ,) ,+ ,'

Why does unraveling occur in a matching market?

´ Proposition 1. Trading cycle with deferred acceptance (or bi- deferred acceptance) (TCDA or BDA) is stable. ´ Implications for DA

´ DA outcome is a subset of TCDA outcome ´ Run M/W-proposing DA in all last large cycles in TCDA è We get M/W-

• ptimal DA results

´ Proposition 3. TCDA/BDA can be used to find all stable equilibria of any two-sided matching mechanism. The number of stable

• utcomes is 2n, where n is the number of the last cycles in

TCDA/BDA with more than 2 agents.

Why does unraveling occur in a matching market?

´ Proposition 3 (Halaburda 2010). The ex-post stable mechanism is Pareto-efficient if and only if it prevents unraveling. ´ Proposition 4 (Ashlagi and Gonczarowski 2016). Acyclic preference

• r priority is a sufficient but not necessary condition for a matching

to be obviously strategy-proof.

Why does unraveling occur in a matching market?

´ Proposition 3 (Halaburda 2010). The ex-post stable mechanism is Pareto-efficient if and only if it prevents unraveling. ´ Proposition 4 (Ashlagi and Gonczarowski 2016). Acyclic preference

• r priority is a sufficient but not necessary condition for a matching

to be obviously strategy-proof. ´ + Proposition 1. ´ Proposition 5. When at least one side has acyclic preference/priority, SD, TTC, IA/BM, DA and TCDA/BDA produce the same unique outcome (regardless it is men-proposing or women- proposing). The matching becomes efficient, stable and (obviously) strategy-proof, and prevents unraveling.

´ Note that, full unraveling can be another unstable equilibrium (see Echenique and Pereyra 2016).

Why does unraveling occur in a matching market?

´ From one-to-one to many-to-one matching ´ Proposition 5. still holds ´ From many-to-one to many-to-one with distributional constraint ´ Proposition 5. fails ´ Proposition 6. In many-to-one matching with distributional constraints and with at least one side has acyclic preference, IA/BM becomes unstable and not strategy-proof. ´ Proposition 7. In many-to-one matching with distributional constraint and with at least one side has acyclic preference, SD, TTC, DA and TCDA produces the same efficient matching outcome. The mechanism is strategy-proof for students, but is manipulable for colleges via capacities.

Application in China College Admission

´ Colleges have tiers and provincial quotas ´ Students submit preferences of limited length ´ Before 2003 ´ Sequential mechanism (Boston mechanism) ´ After 2003 ´ Parallel mechanism (massive reform around 2008) ´ After 2015 ´ DA (parallel mechanism without tiers)

Application in China College Admission

´ Since colleges have identical preferences towards students, ´ Proposition 6 can explain why early admission did not occur after 2015 ´ Proposition 7 can explain why early admission occurred during 2003

• 2015 when the parallel mechanism was used

Application in China College Admission

´ How do we explain the collusion between 2008-2015?

´ It is not because the matching is not group-strategy-proof. ´ It’s because of competition: colleges have identical preferences towards top students, while students have their own capacity constraint such that they cannot participate in all provinces’ decentralized admission.

Application in China College Admission

´ Remaining Question 1: Why did early admission did not occur before 2003?

´ 2 explanations: ´ In the first round of BM, colleges consider students who rank them first. In early admission, only the students who are strong enough and who really like the colleges will apply, and then sign early contract. This process is the same as the first round in BM. ´ Since BM is not strategy-proof for students, and students can strategize by misreporting preferences. Because of acyclic priority, this also benefits colleges, which left them with little incentive to do manipulation. ´ Experiments to test this?

Application in China College Admission

´ How do we prevent unraveling in China’s college admission?

´ Eliminating provincial quotas to eliminate blocking pairs ´ Recruiting students based on one standard ´ Run any matching mechanism (IA, DA, SD, …) ´ We will have an unstable full unraveling equilibrium, but it vanishes easily ´ Or: ´ Preserving students’ and colleges’ preference heterogeneity ´ Run TCDA/BDA ´ Random priority within the last “large” cycles è (obvious) strategy proofness, stability and efficiency

´Thank you! ´Q&A