1. Random matching markets Itai Ashlagi

2. Stable Matchings = Core Allocations

3. Multiplicity  Core has a lattice structure and can be large (Knuth)  Roth, Peranson (1999) – small core in the NRMP  Hitsch, Hortascu, Ariely (2010) – small core in online dating  Banerjee et al. (2009) – small core in Indian marriage markets

4. Random matching markets

5. Questions

6. Large core in random balanced markets

7. Random markets with short preference lists have a small core

8. Literature (multiplicity)

9. Literature  Hitsch, Hortascu, Ariely (2010) – online dating  Banerjee, Duflo, Ghatak, Lafortune (2009) - Indian marriage markets  Abdulkadiroglu, Pathak, Roth (2005) – NYC school choice All use Deferred Acceptance (Gale & Shapely) to make predictions…  Crawford (1991) – comparative statics on adding men (women), but only in a given stable matching.

10. Large core in balanced markets

13. Matching markets are very competitive When there are unequal number of men and women:  The short side is much better off under all stable matchings; roughly, the short side “chooses” and the long side gets “chosen”  sharp effect of competition despite heterogeneity  The core is small; there is little difference between the MOSM and the WOSM  Small core despite long lists and uncorrelated preferences Question: Are there any real/natural matching markets with large cores?

17. Theorem [Ashlagi, Kanoria, Leshno 2013]

18.  That is, with high probability under all stable matchings  Men do almost as well as they would if they choose in a random order, ignoring women’s preferences.  Women are either unmatched or roughly getting a randomly assigned man.  The core is small

19. Corollary 1: One women makes a difference

20. Corollary 2: Large Unbalance

21. Strategic implications

22. Intuition

23. More intuition

24. Proof overview Calculate the WOSM using:  Algorithm 1: Men-proposing Deferred Acceptance gives MOSM  Algorithm 2: MOSM → WOSM Both algorithms use a sequence of proposals by men Stochastic analysis by sequential revelation of preferences.

25. Algorithm 1: Men-proposing DA (Gale & Shapley)

26. Algorithm 2: MOSM → WOSM

28. Illustration of Algorithm 2: MOSM → WOSM

32. New stable match found (Convince yourself that there is no blocking pair), Update match and continue.

33. Illustration of Algorithm 2: MOSM → WOSM

45. Algorithm 2: MOSM → WOSM

47. Proof idea:  Analysis of MPDA is similar to that of Pittel (1989)  Coupon collectors problem  Analysis of Algorithm 2: MOSM → WOSM more involved.  S grows quickly (set of woman that are already matched to best stable partner)  Once S is large improvement phases are rare  Together, in a typical market, very few agents participate in improvement cycles.

48. Why does the set S (women matched to best stable partners) grows quickly?

49. Further questions

52. Men’s average rank of wives

53. Percent of men with multiple stable matches

54. Large Simulations Men's rank under% Men withMen's rank under% Men with MOSMWOSMMultiple StableMOSMWOSM Multiple Stable 101.98 (0.45)2.29 (0.60)13.84 (18.82)1.31 (0.20)1.33 (0.21)1.19 (5.13) 1004.09 (0.72)4.89 (1.08)15.16 (12.98)2.55 (0.26)2.61 (0.27)2.30 (3.15) 1,0006.47 (0.79)7.44 (1.28)11.9 (10.17)4.59 (0.30)4.69 (0.31)1.95 (2.03) 10,0008.80 (0.79)9.80 (1.30)9.45 (8.30)6.88 (0.30)6.98 (0.32)1.46 (1.47) 100,00011.11 (0.83)12.09 (1.31)7.66 (6.60)9.16 (0.31)9.26 (0.32)1.08 (1.02) 1,000,00013.40 (0.80)14.41 (1.27)6.62 (6.04)11.46 (0.30)11.56 (0.32)0.85 (0.80)

55. Summary Random unbalanced matching markets are very competitive:  The short side chooses in all stable matchings  The core is small – most agents have a single stable partner Do matching markets generically have small cores?

56. Recent developments and future directions 1. Why do we observe short preference lists in practice? 2. Efficiency vs Stability (Lee & Yairv 14, Che & Tricieux 14 – consider cardinal utilities) 3. Surplus in random markets with transfers (Romm & Hassidim 2014 – law of one price in random Shapley Shubik model)

57. Topics  Unbalanced matching markets  Matching markets with couples 57

58. The Deferred Acceptance Algorithm [Gale- Shapley’62] Doctor-proposing Deferred Acceptance: While there are no more applications  Each unmatched doctor applies to the next hospital on her list.  Any hospital that has more proposals than capacity rejects its least preferred applicants. 58

59. Source: https://www.aamc.org/download/153708/data/charts1982to2011.pdf 59

60. Two-body problems  Couples of graduates seeking a residency program together. 60

61.  In the 1970s and 1980s: rates of participation in medical clearinghouses decreases from ~95% to ~85%. The decline is particularly noticeable among married couples.  1995-98: Redesigned algorithm by Roth and Peranson (adopted at 1999) Decreasing participation of couples 61

62. Couples’ preferences  The couples submit a list of pairs. In a decreasing order of preferences over pairs of programs – complementary preferences!  Example: 62 AliceBob NYC-ANYC-X NYC-ANYC-Y Chicago-AChicago-X NYC-BNYC-X No MatchNYC-X

63. Couples in the match (n≈16,000) Source: http://www.nrmp.org/data/resultsanddata2010.pdf 63

64. No stable match [Roth’84, Klaus-Klijn’05] 64 C 1212 1 ACAC 2 CBCB B A 1 2

65. Option 1: Match AB 65 C 1212 1 ACAC 2 CBCB C-2 is blocking B A 1 2 B A

66. Option 2: Match C2 66 C 1212 1 ACAC 2 CBCB C-1 is blocking B A 1 2 C 1212

67. Option 3: Match C1 67 C 1212 1 ACAC 2 CBCB AB-12 is blocking B A 1 2 C 1212

68. Stability with couples But:  In the last 12 years, a stable match has always been found.  Only very few failures in other markets. 68

69. Large random market  n doctors, k=n 1- ε couples  λ n residency spots, λ >1  Up to c slots per hospital  Doctors/couples have uniformly random preferences over hospitals (can also allow “fitness” scores)  Hospitals have arbitrary responsive preferences over doctors. 69

70. Stable match with few couples  Theorem [Kojima,Pathak, Roth 10]: In a large random market with n doctors and n 0.5- ε couples, with probability → 1 a stable match exists truthfulness is an approximated Bayes-Nash equilibrium 70 Remark: Kojima-Pathak-Roth actually model this when there are n doctors and n slots and each doctor has a short preference list

71. Theorem [Ashlagi, Braverman, Hassidim 2012]: In a large random market with at most n 1- ε couples, with probability → 1:  a stable match exists, and we find it using a new Sorted Deferred Acceptance (SoDA) algorithm  truthfulness is an approximated Bayes-Nash equilibrium Existence in large markets when the number of couples is not too large

72. The main idea of the proof  We would like to run deferred acceptance in the following order:  singles;  couples: singles that are evicted apply down their list before the next couple enters.  If no couple is evicted in this process, it terminates in a stable matching. 72

73. What can go wrong?  Alice evicts Charlie.  Charlie evicts Bob.  H 1 regrets letting Charlie go. 73 C 1212 1 ACAC 2 CBCB B A 1 2

74. Solution 74 Find some order of the couples so that no previously inserted couples is ever evicted.

75. The couples (influence) graph  Is a graph on couples with an edge from AB to DE if inserting couple AB may displace the couple DE. 75 B A 1 2 C 1212 B A

76. The couples graph 76 AB C D EF G A B EF

77. The couples graph 77 AB C D EF G AB EF

78. The SoDA algorithm  The Sorted Deferred Acceptance algorithm looks for an insertion order where no couple is ever evicted.  This is possible if the couples graph is acyclic. 78 A B CD E F GH

79.  Insert the couples in the order: AB, CD, EF, GH or AB, CD, GH, EF 79 AB CD EF GH

80. Sorted Deferred Acceptance (SoDA) Set some order π on couples. Repeat:  Deferred Acceptance only with singles.  Insert couples according to π as in DA:  If AB evicts CD: move AB ahead of CD in π. Add the edge AB → CD to the influence graph.  If the couples graph contains a cycle: FAIL  If no couple is evicted: Fantastic! 80

81. Couples graph is acyclic  The probability of a couple AB influencing a couple CD is bounded by (log n) c /n≈1/n.  With probability → 1, the couples graph is acyclic. 81

82. Influence trees and the couples graph If: 1.(h,d’)  IT(c j,0) 2.(h,d)  IT(c i,0) 3.Hospital h prefers d to d’ cici cjcj IT(c i,0) - set of hospitals-doctor pairs c i can affect if it was inserted as the first couple cjcj cici hdd’hdd’ IT(c i,r) - similar but allow r adversarial rejections (to capture that other couples may have already applied)

83. Influence trees and the couples graph To capture that other couples have already applied we “simulate” rejections: IT(c i,r) - similar but allow r adversarial rejections Construct the couples graph based on influence trees with r=3/ 

84. Influence trees and the couples graph The influence tree of c consists of all the hospitals- doctors that are likely to be part of the rejection chain due to c ‘s presence. Key steps: 1.Influence tree of each couple is “small”. 2.There are no directed cycles in the couples graph. 3.If c influences some hospital h, then h will belong to that influence tree.

85. Linear number of couples Theorem [Ashlagi, Braverman, Hassidim 12]: in a random market with n singles, α n couples and large enough λ >1, with constant probability no stable matching exists. Idea: 1. Show that a small submarket with no stable outcome exists 2. No doctor outside the submarket ever enters a hospital in this submarket market

86. Results from the APPIC data  Matching of psychology postdoctoral interns.  Approximately 3000 doctors and 20 couples.  Years 1999-2007.  SoDA was successful in all of them.  Even when 160 “synthetic” couples are added. 86

87. SoDA: the couples graphs  In years 1999, 2001, 2002, 2003 and 2005 the couples graph was empty. 87 2008200420062007

88. number of doctors SoDA: simulation results  Success Probability(n) with number of couples equal to  n. 4% means that ~8% of the individuals participate as couples. 88 808 per 16,000 ≈ 5% probability of success

89. Summary and some directions for research 1. More structure on couples preferences (some “cities” structure is given in Ashlagi, Braverman, Hassidim). Relax preferences of hospitals. 2. What to do if there is no stable matching? Roth and Peranson make decisions in the algorithm when there is no stable matching. 3. What would be a good strategy for an employer in a big city? In a rural area?