Matching and Market Design Theory and Practice Xiang Sun May 12, 2017 ii Contents Acknowledgement vii 1 Introduction 1 1.1 Matching and market design .......................................... 1 1.2 Time line of the main evolution of matching and market design ....................... 2 I Two-sided matching 9 2 Marriage 11 2.1 The formal model ............................................... 11 2.2 Stability and optimality ............................................ 12 2.3 Deferred acceptance algorithm ........................................ 14 2.4 Properties of stable matchings I ........................................ 16 2.5 Properties of stable matchings II ........................................ 21 2.6 Extension: Extending the men’s preferences ................................. 24 2.7 Extension: Adding another woman ...................................... 26 2.8 Incentive compatibility I ............................................ 29 2.9 Incentive compatibility II ........................................... 35 2.10 Non-bossiness ................................................. 38 3 College admissions 41 3.1 The formal model ............................................... 41 3.2 Stability ..................................................... 42 3.3 The connection between the college admissions model and the marriage model .............. 44 3.4 Deferred acceptance algorithm and properties of stable matchings ..................... 45 i Contents ii 3.5 Further results for the college admissions model ............................... 49 3.6 Incentive compatibility ............................................. 52 3.6.1 Preference manipulation ....................................... 53 3.6.2 Capacity manipulation ........................................ 54 3.7 Comparison of marriage problems and college admissions ......................... 56 3.8 National intern matching program ...................................... 57 3.9 New York City high school match ....................................... 59 II One-sided matching 61 4 Housing market 63 4.1 The former model ............................................... 63 4.2 Top trading cycles algorithm .......................................... 66 4.3 Incentive compatibility ............................................. 72 4.4 Axiomatic characterization of top trading cycles algorithm ......................... 77 5 House allocation 81 5.1 The former model ............................................... 81 5.2 Simple serial dictatorship and core from assigned endowments ....................... 82 5.3 Incentive compatibility ............................................. 87 5.4 Neutrality .................................................... 88 5.5 Consistency ................................................... 91 5.6 Random house allocation ........................................... 92 6 House allocation with existing tenants 95 6.1 The former model ............................................... 95 6.2 Real-lief mechanisms .............................................. 96 6.2.1 Random serial dictatorship with squatting rights .......................... 96 6.2.2 Random serial dictatorship with waiting list ............................. 97 6.2.3 MIT-NH4 mechanism ......................................... 98 6.3 Top trading cycles algorithm .......................................... 99 6.4 You request my house—I get your turn algorithm .............................. 104 6.5 Axiomatic characterization of YRMH-IGYT ................................. 109 6.6 Random house allocation with existing tenants ................................ 110 7 Hierarchical exchange rule 111 Contents iii 8 Random assignment mechanism 113 8.1 Random assignment problem ......................................... 113 8.2 Random priority mechanism ......................................... 115 8.3 Simultaneous eating algorithm and probabilistic serial mechanism ..................... 116 8.4 Efficiency .................................................... 119 8.4.1 Basics .................................................. 119 8.4.2 Ordinal efficiency ........................................... 120 8.4.3 Efficiency of RP and PS ........................................ 125 8.5 Fairness ..................................................... 127 8.5.1 Anonymity ............................................... 127 8.5.2 Envy-freeness ............................................. 129 8.5.3 Equal treatment of equals ....................................... 133 8.6 Incentive compatibility ............................................. 133 8.7 RP vs PS .................................................... 136 8.8 Impossibility results .............................................. 137 8.9 Large markets .................................................. 137 8.10 Implementing random assignments ...................................... 137 III School choice 139 9 Introduction to school choice 141 9.1 The former model ............................................... 141 9.2 Boston school choice mechanism (immediate acceptance mechanism) ................... 145 9.3 Deferred acceptance algorithm and student-optimal stable mechanism ................... 147 9.4 Top trading cycles mechanism ......................................... 150 9.5 Case study: Chinese college admissions .................................... 152 10 Acyclicity 155 10.1 Cycles and efficiency of deferred acceptance algorithm ............................ 155 10.2 Robust stability ................................................. 161 11 Efficiency improvement on student-optimal stable mechanism 165 11.1 Efficiency-adjusted deferred acceptance algorithm .............................. 165 11.2 Simplified efficiency-adjusted deferred acceptance algorithm ........................ 172 11.3 Stable improvement cycle algorithm ...................................... 177 Contents iv 12 School choice with weak priorities 179 12.1 Weak priorities ................................................. 179 12.2 DA with tie breaking rules ........................................... 180 12.3 Stable improvement cycles algorithm ..................................... 181 13 Affirmative action 185 13.1 The formal model ............................................... 185 13.2 Affirmative action policies with majority quotas ............................... 186 13.3 Affirmative action policies with minority reserves .............................. 189 IV Kidney exchange 193 14 Kidney exchange I 195 14.1 Background ................................................... 195 14.2 The model ................................................... 197 14.3 Multi-way kidney exchanges with strict preferences ............................. 198 15 Kidney exchange II 207 15.1 Two-way kidney exchanges with 0–1 preferences ............................... 207 15.2 Multi-way kidney exchanges with 0–1 preferences .............................. 207 15.3 Dynamic kidney exchange ........................................... 207 15.4 Lung exchange ................................................. 207 V Topics 209 16 The assignment game 211 16.1 The core of the game .............................................. 211 16.2 The formal model ............................................... 211 16.3 The core of the assignment game ....................................... 211 17 Matching with transfers 213 17.1 The substitutable preferences ......................................... 213 17.2 A model with money and gross substitutable preferences .......................... 215 18 Matching with constraint 219 19 Matching with incomplete information 221 Contents v Bibliography 223 Contents vi Acknowledgement This note was initiated when I gave lectures at Wuhan University in Spring 2014. I thank Wei He, Gaoji Hu and Qian- feng Tang for helpful discussion. This note is much influenced by the following books, surveys, and lecture notes • Atila Abdulkadiroğlu and Tayfun Sönmez, Matching Markets: Theory and Practice, in Advances in Economics and Econometrics Theory and Applications, Volume II, Tenth World Congress. • Fuhito Kojima, Lecture notes on Market Design, 2015. Available at Kojima’s homepage. • Alvin E. Roth’s blog on Matching and Market Design. • Alvin E. Roth and Marilda A. Oliveira Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Cambridge University Press, 1992. • Tayfun Sönmez, Mini-Course on Matching. Available at Sönmez’s homepage. • Tayfun Sönmez and M. Utku Ünver, Matching, Allocation, and Exchange of Discrete Resources, in Handbook of Social Economics, Volume 1A (Jess Benhabib, Alberto Bisin and Matthew O. Jackson Eds.), Elsevier B.V., 2010. • Qianfeng Tang and Yongchao Zhang, Lecture notes on matching, 2015. • Jerusalem Summer School in Matching and Market Design (with recorded lectures), 2014. Available at http: //www.as.huji.ac.il/schools/econ25. I am grateful for the following students who provide lots of helpful suggestions and comments: Xiaoguang Chen (陈 晓光), Yichen Chen (陈奕辰), Jun Cheng (成军), Yongkang Hu (胡永康), Liucong Huang (黄柳淙), Yikuan Huang (黄 怡宽), Zhijiang Huang (黄志疆), Yiwen Jin (金怡文), Jia’en Li (李佳恩), Qin Li (李擎), Xiao Lin (林潇), Haomin Liu (刘昊旻), Yuting Liu (刘雨婷), Jie Lu (陆劼), Xiaojie Mao (毛小介), Keji Mei (梅珂佶), Tianchen Song (宋天辰), Xunkang Tian (田珣康), Rui Wang (汪瑞), Xiao Wang (王筱), Ya Wen (文雅), Kaiwen Wu (吴锴文), Yue Wu (伍玥), Weihao Yan (闫玮昊), Chen Zhang (张晨), Ying Zhou (周颖). vii viii Chapter 1 Introduction Contents 1.1 Matching and market design ......................................