Game theorists are special people
• Lots of people came to Brazil for the World Cup
• We came to Brazil to celebrate Marilda, and her career, and the area in which she has worked and made famous.
Some quick personal history • I first met Marilda in 1985, in PiSsburgh: • From her CV: VisiUng PosiUons: • Department of Economics, University of PiSsburgh, U.S.A. -‐ 03/1985 to 07/1985, 01/1986 to 02/1986, 06/1986 to 07/1986, 06/1987 to 07/1987, 06/1988 to 07/1988, 08/1989 to 12/1989, 08/1991, 08/1992, 08/1996 to 05/1997 .
Many important results in matching • Existence of stable matchings
– Sotomayor: simple matchings, non-‐construc3ve proof • OpUmal stable matchings—lacces, other structure
– Sotomayor, many papers on lacce structure – Gale and Sotomayor, decomposi3on lemma
• IncenUves: Demange Gale and Sotomayor on – limits of manipula3on – Equilibria – blocking lemma – Simultaneous mul3-‐item auc3ons
• Unmatched workers and firms—rural hospitals – Gale and Sotomayor: early general results for college admissions problem
• Couples – Sotomayor: counter example
• Many to one matching has different structure than one to one – Sotomayor and Roth: structure of stable matchings in college admissions
problem
Two kinds of fundamental theoreUcal results
• Some theory is fundamental because it plays a big role in producing new theory, and understanding mathemaUcal structure
• And some theory is fundamental because it helps us understand the world, make sense of empirical observaUons, and explains how and why successful market designs work.
• Fundamental results don’t have to be old results, but it’s easier to tell which results are fundamental by giving them some Ume to show their worth.
Some papers with fundamental results • Some Remarks on the Stable Matching Problem, David Gale and Marilda Sotomayor, Discrete Applied Mathema1cs, 1985 (decomposiUon)
• Mul3-‐Item Auc3ons, Gabrielle Demange, David Gale, and Marilda Sotomayor, Journal of Poli1cal Economy, 1986 (simultaneous ascending aucUon)
• A Further Note on the Stable Matching Problem, Gabrielle Demange, David Gale, and Marilda Sotomayor, Discrete Applied Mathema1cs, 1987 (limits to manipulaUon)
• A Non-‐construc3ve Elementary Proof of the Existence of Stable Marriages, Marilda Sotomayor, Games and Economic Behavior, 1996 (structure, sUll to come?)
Mul3-‐Item Auc3ons By Gabrielle Demange, David Gale, and Marilda Sotomayor Journal of Poli1cal Economy, 1986 “A collecUon of items is to be distributed among several bidders, and each bidder is to receive at most one item. Assuming that the bidders place some monetary value on each of the items, it has been shown that there is a unique vector of equilibrium prices that is opUmal, in a suitable sense, for the bidders. In this paper we describe two dynamic aucUon mechanisms: one achieves this equilibrium and the other approximates it to any desired degree of accuracy.”
In the United States, the Federal CommunicaUons Commission (FCC) conducts aucUons of licenses for electromagneUc spectrum. The FCC has been conducUng compeUUve aucUons since 1994 … the FCC has conducted 87 spectrum aucUons, which raised over $60 billion for the U.S. Treasury... The aucUon approach is widely emulated throughout the world. The FCC auc3ons have used a Simultaneous Mul3ple Round Auc3on … (SMRA, also referred to as the Simultaneous Ascending Auc3on) in which groups of related licenses are auc3oned simultaneously over many rounds of bidding.. …As the auc3on progresses, the standing high bid changes to highest new bid and the corresponding bidder is the person who makes said bid. In addi3on to pos3ng the round results, minimum bids for the next round are also posted. … For an aucUon to come to a close there are several different opUons. McAfee, suggested that aucUons should come to a close aler a predetermined number of rounds, in which the license receives no new bids. Wilson and Paul Milgrom of Stanford University proposed that all auc3ons should end simultaneously, when there is no new bid on a license. To date, the laSer is used in the spectrum aucUons.
Spectrum AucUons (from Wikipedia:)
Some Remarks on the Stable Matching Problem, David Gale and Marilda Sotomayor Discrete Applied Mathema1cs, 1985
DecomposiUon
• Implies that median stable matchings exist • Applies to one-‐sided roommate problems…
Theorem 4.11: Limits on successful manipulaUon (Demange, Gale, and Sotomayor). Let P be the true preferences (not necessarily strict) of the agents, and let P differ from P in that some coaliUon C of men and women misstate their preferences. Then there is no matching µ, stable for P, which is preferred to every stable matching under the true preferences P by all members of C
The limits to manipulaUon theorem sheds new light on an empirical puzzle • Unstable matching mechanisms are olen observed to fail—for reasons that seem closely related to producing unstable matchings, or giving agents incenUves to misrepresent (or both)
• Stable matching mechanisms have mostly succeeded—for reasons that seem closely related to producing a stable matching
• But no stable mechanism makes it a dominant strategy for all par3cipants to state their true preferences
20
Stable Clearinghouses (those now using the Roth Peranson Algorithm)
NRMP / SMS: Medical Residencies in the U.S. (NRMP) (1952) Abdominal Transplant Surgery (2005) Child & Adolescent Psychiatry (1995) Colon & Rectal Surgery (1984) Combined Musculoskeletal Matching Program (CMMP) • Hand Surgery (1990) Medical SpecialUes Matching Program (MSMP) • Cardiovascular Disease (1986)
• Gastroenterology (1986-‐1999; rejoined in 2006)
• Hematology (2006) • Hematology/Oncology (2006) • InfecUous Disease (1986-‐1990; rejoined in 1994) • Oncology (2006) • Pulmonary and CriUcal Medicine (1986) • Rheumatology (2005) Minimally Invasive and GastrointesUnal Surgery (2003) Obstetrics/Gynecology • ReproducUve Endocrinology (1991) • Gynecologic Oncology (1993) • Maternal-‐Fetal Medicine (1994) • Female Pelvic Medicine & ReconstrucUve Surgery (2001) Ophthalmic PlasUc & ReconstrucUve Surgery (1991) Pediatric Cardiology (1999) Pediatric CriUcal Care Medicine (2000) Pediatric Emergency Medicine (1994) Pediatric Hematology/Oncology (2001) Pediatric Rheumatology (2004) Pediatric Surgery (1992)
Primary Care Sports Medicine (1994) Radiology • IntervenUonal Radiology (2002) • Neuroradiology (2001) • Pediatric Radiology (2003) Surgical CriUcal Care (2004) Thoracic Surgery (1988) Vascular Surgery (1988) Postdoctoral Dental Residencies in the United States • Oral and Maxillofacial Surgery (1985) • General PracUce Residency (1986) • Advanced EducaUon in General DenUstry (1986) • Pediatric DenUstry (1989) • OrthodonUcs (1996) Psychology Internships in the U.S. and CA (1999) Neuropsychology Residencies in the U.S. & CA (2001) Osteopathic Internships in the U.S. (before 1995) Pharmacy PracUce Residencies in the U.S. (1994) ArUcling PosiUons with Law Firms in Alberta, CA(1993) Medical Residencies in CA (CaRMS) (before 1970)
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BriUsh (medical) house officer posiUons • Edinburgh (1969) • Cardiff (197x) New York City High Schools (2003) Boston Public Schools (2006)
Large core in balanced markets
Theorem[PiSel 1998, Knuth, Motwani, PiSel 1990] Consider a random market with 𝒏 men and 𝒏 women. 1. With high probability, the average ranks in the MOSM are
𝑅↓Men =log 𝑛 and 𝑅↓Women = 𝑛/log𝑛 2. The frac3on of agents that have mul3ple stable partners tends to 1 as 𝒏 tends to infinity.
Theorem [Ashlagi, Kanoria, Leshno 2013]
Consider a random market with 𝑛 men and 𝑛+𝑘 women, for 𝑘≤𝑛/2. With high probability in any stable matching
𝑅↓Men ≤1.01 log(𝑛/𝑘 ) and 𝑅↓Women ≥ 𝑛/1.01 log(𝑛/𝑘) Moreover, 𝑅↓Men (WOSM)≤(1+𝑜(1)) 𝑅↓Men (MOSM) 𝑅↓Women (WOSM)≥(1−𝑜(1)) 𝑅↓Women (MOSM)
And 𝒐(𝒏) of men and women have mul3ple stable matches.
Percent of matched men with mulUple stable partners |𝒲|=40
0
10
20
30
40
50
60
70
20 25 30 35 40 45 50 55 60 65 70 75 80
Average Percent of M
atched
Men
with
Mul3p
le Stable Pa
rtne
rs
Number of Men
So the set of stable matchings is mostly small
• So the limit on manipulaUon theorem says that virtually no one can do beSer than to state their true preferences – In which case stable mechanisms will produce stable matchings…
Structure of set of stable matchings
• Lacces – Tarski’s theorem: works on the lacce of pre-‐matchings, fixed points are the sublacce of stable matchings
– Preferences are subsUtutes— – MulUtude of subsUtute condiUons (starUng with Kelso and Crawford)
– Maybe not the right unifying structure? • Other possibiliUes – DecomposiUon?