Introduction to Matching Theory and Market...

Introduction to Matching Theory and Market Design

by Fuhito Kojima (Stanford)

NUS, July 2018

Introduction

This is a short tutorial about matching theory and its application to market design.

I will quickly cover basic theory first, and then move to recent developments.

Part 0: What is matching theory? What is market design?

c.f. Roth, 2015, “Who Gets What and Why: The new Economics of

Matchmaking and Market Design”

What are markets?

A common perception: Economists study “markets”

What are markets?

Commodity markets:

Supply meets demand, guided by price

Matching markets

In some markets, prices don’t work perfectly

Matching markets is a prime example: you care about who you are interacting with.

You cannot just choose who to match with. You should be chosen!

Matching markets examples

You cannot just pay to get into NUS—you should be admitted

You cannot just decide to work for Google—you should be hired!

Market Design

In matching markets, prices don’t do all the job.

Instead, market organizers (“matchmakers”) make rules to facilitate good matching; market design

Labor markets for medical residents/interns (NRMP)

school choice

organ transplantation

daycare (nursery schools)

Questions

In matching markets, what kind of systems work well?

What does “work well” even mean?

This lecture tries to answer these questions

Gale and Shapley, “College Admissions and the Stability of

Marriage” American Mathematical Monthly, 1962

Roth, “The Evolution of the Labor Market for Medical

Interns and Residents: A Case Study in Game Theory” JPE,

Part 1: Standard Model

based on (mostly)

Standard model: setting

Doctors (1,2,…,i,j,…) and hospitals (A,B,…)

Many-to-one matching

Preferences over each other (& outside option ∅)

Matching; specify who matches with whom

(students/workers) (schools/firms)

Standard Model: Stability

A matching is stable if

Individually rational

No blocking pairs

Standard Model: Stability

A matching is stable if

Individually rational

No blocking pairs

Interpretations

encourage participation, prevent loopholes

Core (so Pareto efficient)

Fairness

(doctor-proposing) Deferred Acceptance Algorithm (“DA”)Step 1:

Doctors apply to their 1st choice hospitals

Each hospital (tentatively) holds its most preferred acceptable applicants up to capacity and rejects the rest

Step t≧2:

Rejected doctors apply to their next choice hospitals

Each hospital combines both existing and new applicants, holds its most preferred acceptable ones up to capacity, and rejects the rest

Repeat until no more applications are made and finalize the match

Existence

The outcome of DA is stable (Gale and Shapley 1962 AMM)

Individual Rationality

No blocking pair

Real-life examples

DA is used in NRMP (Roth 1984 JPE)

Other markets; UK medical match (Roth 1991 AER), Turkish college admission (Balinski and Sonmez 1999 JET), UK schools (Pathak and Sonmez 2013 AER)…

Intentional market design adopts DA; UK medical match, JRMP, APPIC, NYC, Boston,…

Incentives

DA is strategy-proof for the doctors (Roth 1982 MOR; Dubins and Freedman 1981 AMM)

DA is “group strategy-proof”

Generalizations: Hatfield and Milgrom (2005 AER), Hatfield and Kojima (2009 GEB, 2010 JET), Hirata and Kasuya (2016 JET), Hatfield, Kominers, and Westkamp (2016)

DA is not strategy-proof

i j A B

A B j i

B A i j

Preference table(each hospital has 1 seat)

Under truth-telling:i j A B

A B j i

B A i j

A B j i

B A i j

A B j i

B A i j

❤❤

A B j i

B A i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

A B j i

B A i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

❤❤

❤💔

A B j i

B A i j

AB i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

❤❤

A B j i

B A i j

When A misreports:

❤❤

❤∅

A B j i

B A i j

When A misreports:Profitable misreport!

❤❤

❤∅

Impossibility Theorem

Theorem (Roth 1982 MOR): There exists no stable and strategy-proof mechanism.

Proof? (C.f. Roth and Sotomayor 1990, p. 88)

So, a “perfect mechanism” is impossible!

Roth and Peranson “The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design”, AER 1999

Kojima and Pathak “Incentives and Stability in Large Two-Sided Matching Markets” AER 2009

Kojima Pathak, and Roth “Matching with Couples: Stability and Incentives in Large Markets” QJE 2013

based on (mostly)

Part 2: Large Markets and “Approximate Market Design”

Why are DAs used?

Puzzle; DA isn’t strategy-proof, but still used widely in practice

Why are DAs used?

Puzzle; DA isn’t strategy-proof, but still used widely in practice

NRMP, JRMP, UK medical match, APPIC, NYC, Boston, …

NRMP dataRoth and Peranson (1999 AER); DA is “approximately IC”: # of hospitals with profitable misreporting in NRMP

Year 1987 1993 1994 1995 1996

# of hospitals 3170 3622 3662 3745 3758

# manipulable 15 12 15 23 14

Simulation

Simulationn doctors, n hospitals (one seat in each hospital)k=#{hospitals acceptable to each doctor}

iid uniform random preferences

p(n)=proportion of hospitals that can profitably manipulate

Simulationn doctors, n hospitals (one seat in each hospital)k=#{hospitals acceptable to each doctor}

iid uniform random preferences

p(n)=proportion of hospitals that can profitably manipulate

Large market →approximate IC(?)

Theory: Setup

Sequence of markets, indexed by # of hospitals, n.

Theory: Setup

k=#{hospitals acceptable to each doctor} constant; “short list” (→Detail)

Theory: Setup

k=#{hospitals acceptable to each doctor} constant; “short list” (→Detail)

Game: each participant submits preference, and DA produces a matching

Theory: ResultTheorem (Immorlica and Mahdian 2005 SODA, Kojima and Pathak 2009 AER); For any Ɛ>0, there is n such that truth-telling is an Ɛ-Bayes-Nash equilibrium for any market with more than n hospitals.

Theory: Result

So a stable mechanism may be approximately incentive compatible in large markets!

Theorem (Immorlica and Mahdian 2005 SODA, Kojima and Pathak 2009 AER); For any Ɛ>0, there is n such that truth-telling is an Ɛ-Bayes-Nash equilibrium for any market with more than n hospitals.

Intuition

IntuitionRecall DA is strategy-proof for doctors.

Reason for profitable manipulations: rejection chain.

A rejected doctor applies to the manipulating hospital

In large markets; Doctors along the rejection chain are likely to apply to a hospital with a vacant position and be accepted.

→ rejection chain stops; no benefit of manipulation.

→Discussion (Lee (2014), Ashlagi et al. (2015))

Existence of stable matchings

Existence of stable matching fails when standard conditions are violated

Couples who want two jobs

Couples who want two jobs(h,w)

(A1,A2)(B1,B2)(C1,C2)(∅,∅)

Couples who want two jobs(h,w)

(A1,A2)(B1,B2)(C1,C2)(∅,∅)

Couples are

5-10% in NRMP

1-2% in APPIC

Nonexistence with couples(h,w) s A B

(A,B) A h s

(∅,∅) B s w

Each hospital has one seat

Nonexistence with couples

Can verify each matching is unstable

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

Failure of DA-like algorithm

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

s❤ 💔

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

hw s❤💔

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

(h,w) s A B

(A,B) A h s

(∅,∅) B s w

Cycle!

Existence in practice

Stable matchings exist for most cases even with couples

NRMP, APPIC…

Why is this?

NRMP, APPIC…

Why is this?

Preference restriction? (→Detail)

Theoretical ResultConsider a large market setting as in Kojima and Pathak (2009), and assume the proportion of couples is small compared to the market size.

Theorem (Kojima et al. 2013 QJE); The probability that a stable matching exists

converges to one as the market size goes to infinity.

Theoretical ResultConsider a large market setting as in Kojima and Pathak (2009), and assume the proportion of couples is small compared to the market size.

Theorem (Kojima et al. 2013 QJE); The probability that a stable matching exists

converges to one as the market size goes to infinity.

So a stable matching existence is robust!�28

Intuition

A DA-like algorithm finds a stable matching if no couples are displaced later.

Intuition

Large markets: rejection chains are likely to stop at a vacant position

Intuition

→ produces a stable matching.

Intuition

→ produces a stable matching.

→Disucussion (Ashlagi et al. 2014 OR)

Large market referencesGeneral preference complementarity: Azevedo-Hatfield (2017), Che-Kim-Kojima (2018 ECMA), Nguyen-Vohra (2016)

Object allocation: Manea (2009 TE), Kojima-Manea (2010, JET), Che-Kojima (2010 ECMA), Lee-Yariv (2015), Liu-Pycia (2013), Azevedo-Budish (2013), Hashimoto (2016), Che-Tercieux (2015 TE), Akbarpour-Nikzad (2015), Mennle-Seuken (2015), Peivandi-Nguyen-Vohra (2015 JET), Ashlagi-Shi (2014, MS),

Other uses: Abdulkadiroglu-Che-Yasuda (2015 AEJ:Micro), Ashlagi-Shi (2015 MS), Arnosti (2015), Azevedo-Leshno (2014 JPE), Hatfield-Kojima-Narita (2016 JET), Echenique-Lee-Shum-Yenmez (2013 ECMA), Echenique-Pereyra (2015 TE), Nei&Pakzad-Hurson (2015), Kennes-Monte-Tumennasan (2015), Peski (2015)

Part 3: Matching with Constraints

Roth, “On the allocation of residents to rural hospitals: a general property of two-sided matching markets” Econometrica 1986

Kamada and Kojima, “Efficient matching under distributional constraints: Theory and applications” AER 2015

based on (mostly)

Overview

Many matching markets are subject to constraints

Medical specialties

Multiple school programs sharing one building

Affirmative action (diversity constraints)

Specific real-life examples: Next

Case study: Japan

Japan residency matching program (JRMP) adopted doctor-proposing deferred acceptance mechanism (DA) in 2003

Case study: Japan

Critics claimed too many doctors are allocated to metropolitan areas under DA.

Case study: Japan

Critics claimed too many doctors are allocated to metropolitan areas under DA.

Government introduced a regional cap as a constraint

Japanese Data on Regional Caps

*** paste the prefecture picture ***

Tokyo$

Osaka$

Kanagawa$

Aichi$

Fukuoka$

Hokkaido

Chiba$

Kyoto$

Hiroshim

a$Okayama$

Nagano$

Miyagi$

Ibaraki$

Okinawa$

Tochigi$

a$Niigata$

Nagasaki$

Fukushim

a$Kagoshim

a$Ishikawa$

Yamaguchi$

$Nara$

Aomori$

Iwate$

Yamagata$

$Wakayam

a$Shiga$

Kagawa$

Yamanashi$

Fukui$

Kochi$

Tokushim

a$Saga$

ToJori$

Miyazaki$

regional$cap$

total$capacity$

Prefecture�

f$PosiNon

Total$Capacity$

$Regional$Cap�

Prefecture�

f$PosiNon

Total$Capacity$

$Regional$Cap�

Total capacity

Regional Cap

Region (prefecture)

More examples of constraints

Chinese graduate school admission

academic/professional programs

College admission in Hungary & Ukraine

state-financed/privately-financed seats

Medical match in U.K. (regional cap)

Teacher assignment in Scotland (regional cap)

Why constraints?

One question: Why not tweak with mechanisms to change doctor allocation pattern without imposing constraints?

The idea was discussed in earlier days of the American medical intern market (Sundarshan and Zisook, 1981):

``The United States suffers a terrible problem of maldistribution of physicians, with urban areas being relatively over-served and inner-city and rural areas being relatively under-served. At present, although approximately 100 hospitals fill every residency position, there are over 100 hospitals that do not receive a single application. This maldistribution would only be worsened by the only rational alternative to the present matching program, the mirror-image program, which favors students.’'

Next theorem will provide some answer…

Why constraints?

Rural Hospital Theorem (Roth 1986): For each hospital that leaves at least one position vacant in a stable matching, the hospital obtains the same set of doctors in every stable matching.

So tweaking mechanisms won’t help unless one relaxes (or gives up) stability in some manner.

Research Goal

Main questions:

Research Goal

Main questions:

Desirable design under constraints?

Research Goal

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Standard model: setting

Doctors (1,2,…,i,j,…) and hospitals (A,B,…)

Many-to-one matching

Preferences over each other (& outside option ∅)

Matching; specify who matches whom

Stability= IR + no blocking pair

Deferred acceptance (DA)

Model of constraints

Standard two-sided matching except

Each hospital belongs to a region

Each region has exogenous regional cap

A matching is feasible if, for each region,

(# of doctors in the region) ≦ (regional cap)

(nonnegative integer)

Japanese mechanism

Japanese mechanism(doctor-proposing) DA:

Begin with an empty matching. Repeat Steps below:

Application part: Each unmatched doctor applies to her favorite hospital that has not rejected her yet (if any).

Acceptance/Rejection part: From both its tentatively matched and new applicants:

Each hospital (tentatively) accepts its favorite acceptable applicants up to its capacity.

Go to Next Step.

Go to Next Step.artificial capacity

- less than hospital’s capacity- adds up to regional cap

Go to Next Step.

JRMP (Japanese mechanism)

artificial capacity- less than hospital’s capacity- adds up to regional cap

JRMP is inefficient and unstable!

Let there be one region, with cap 10.

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

Each hospital has 10 seats

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

JRMP (target capacity=5 each):

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

12 34 5 6 7 8

9 10�42

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 3 4 5 6 7 8

9 10�42

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 56 7 8

9 10�42

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 56 7 8

9 10💔💔�42

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 56 7 8

9 10�42

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 56 7 8

9 10inefficient & unstable!�42

New mechanism

New mechanismJapanese mechanism:

Each hospital (tentatively) accepts its favorite acceptable applicants up to its artificial capacity.

Go to Next Step.

With an exogenous order, hospitals take turns to (tentatively) accept favorite remaining applicants until (i) the regional cap becomes full or (ii) no doctor remains to be processed. Go to Next Step.

Flexible DA (FDA)

FDA exampleLet there be one region, with cap 10.

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

FDA (artificial capacity=5 each):

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 3 4 5 6 7 8 9 10

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 3 4 5 6 7 8 9 10

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 5 6 7 8 9 10

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 5 6 7 8 9 10

❤❤❤❤❤�44

1st phase

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 5 6 7 8 9 10

💛💛❤❤❤❤❤�44

1st phase2nd phase

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 34 5 6 7 8 9 10

💛💛efficient & stable!

❤❤❤❤❤�44

1st phase2nd phase

FDA theoretical results

FDA outcome is (constrained) Pareto efficient and “stable”

“stability under constraints” needs to be defined carefully; later

FDA is strategy-proof for doctors

Every doctor weakly prefers FDA to JRMP

So the set of matched doctors increases

Simulation: number of matched/unmatched doctors

DA" JRMP" FDA"

ber'of'doctors'

Mechanism'

Unmatched"doctors"

Matched"doctors"

Matched

Unmatched

unconstrained

DAFDAJRMP

Simulation: Rank distributions

1# 2# 3# 4# 5# 6# 7# 8# 9#

ber'of'doctors'

Ranking'of'the'matched'hospital'

DAFDAJRMP

Ranking of matched hospitals

Simulation: Doctor distribution

y"="0.0001x"("0.0211"R²"="0.00327"

100" 300"

%JRMP)/JRM

# of doctors/100,000 persons

Conclusion

Matching theory has been applied to many markets in practice

Conclusion

Classical theory may not directly apply to markets in practice, suggesting new kinds of theory

Conclusion

Large market and “approximate market design”

Conclusion

Matching with constraints

Conclusion

Interaction with markets in practice continues to enrich theory.

Fair Matching under Constraints(with application to daycare allocation)

Fuhito Kojima (joint with Yuichiro Kamada)

Daycare Allocation

• Matching mechanisms are used for daycare allocation.

• Policy experiments to improve matching mechanisms.

• Teacher-child ratio (& space-child ratio) varies with age (Okumura, 2016)

→not capacity; instead, matching with constraints

Daycare seats over-demand in Japan

Markets with constraints• Many other matching markets are subject to

constraints too

• Affirmative action (diversity constraints)

• Gender composition in workplace

• More real-life examples (later)

• Question: Desirable outcomes & mechanisms?

Main Results• Stable matching does not always exist

• Fair matchings are characterized via fixed points of a function

• Necessary and sufficient condition for existence of a student-optimal fair matching (SOFM)

• general upper-bound

• Application to daycare allocation with data

Model• Students (denoted i, I) and schools (denoted s, S)

• Many-to-one matching

• Each Student has strict preferences over schools (& outside option, ∅)

• Each school has a strict priority order over students

• Generalizable to weak priority (i.e., ties)

Constraints• Each school s is subject to a constraint

• For each subset I’ of students, a constraint tells “feasible” or “infeasible”

• c.f. Constraints at the level of sets of schools (Biro et al. 2010, Kamada and Kojima, 2015, 2016a,b, Kojima et al. 2016, Goto et. al 2016)

• For each school, assume there is at least one feasible set of students.

Desirable properties• Feasibility (students feasible at every school), IR (students should

be matched to ∅ or better)

• Non-wastefulness: there are no i, s, such that

• i prefers s to her own assignment,

• moving i to s results in a feasible matching

• Fairness (elimination of justified envy): there are no i, i’, s, such that

• i’ is matched to s and i has a higher priority than i’ at s

Discussion on fairness• Fairness (elimination of justified envy): there are no i, i’, s, such that

• i’ is matched to s and i has a higher priority than i’ at s

• Appropriate fairness concept depends on applications

• Labor markets (medical match): weak fairness

• College admission with disability, disaster relief material: fairness

• Non-existence problem robust to fairness concepts employed

• and replacing i’ with i is feasible at s

Weak fairness

Preliminary Facts

• Fact 1: feasibility & IR & fairness & non-wastefulness ⇔ stability

• Fact 2: Stability (=Feasibility & IR & Fairness & Non-wastefulness) leads to non-existence

• “Necessary and sufficient” condition turns out to be capacity constraints (later)

Fair matching

• Approach: Don’t insist on (exact) non-wastefulness but require fairness (+ feasibility, IR)

• Existence? Structure?

• Characterization via a mapping

Cutoff adjustment function• Ps : the cutoff (=lowest priority/“score” to be admitted) at school s;

• regarded as an element in {1,…,n,n+1}, where n:=number of students.

• P=(Ps)s: a cutoff profile at all schools.

• D(P)=(Ds(P))s: the demand profile at P • each student chooses favorite available school given P (or ∅)

• Cutoff adjustment function T from cutoff profiles to themselves: • Ts(P)=Ps+1 (mod n+1) if Ds(P) is infeasible (i.e., “over-demanded”) • Ts(P)=Ps otherwise.

• T is like Walrasian tatonnement but doesn’t try to eliminate under-demand

Characterization

• Proof: Given P induces matching D(P)=(Ds(P))s,

• there is no guarantee that D(P) is feasible, but

• D(P) is IR and fair

• P=T(P) iff D(P) is feasible by definition of T. 61

Theorem: If a cutoff profile P is a fixed point of T, then the induced matching is feasible, individually rational, and fair. Moreover, if a matching is feasible, individually rational, and fair, then there exists a cutoff profile that induces it.

Problem with fairness

• An arbitrary fair matching may be undesirable.

• Is there a “(most) desirable” fair matching?

SOFM• A matching is a student-optimal fair matching (SOFM) if

1. fair, IR, feasible, and

2. weakly preferred by every student to any matching satisfying (1).

• Similar to “student-optimal stable matching” in standard case

• note a stable matching may not exist

General upper bound

• We say constraints are general upper-bound if every subset of a feasible subset is also feasible • subsume standard settings like (1) capacity

constraints and (2) type-specific quotas (diversity in schools), but exclude e.g., minimum (floor) constraints

• More examples of general upper-bound; next

General upper bound• Recall general upper-bound; every subset of a

feasible subset is also feasible • More (less standard) examples of general upper-bound

• College admission with students with disability (budget constraint)

• Refugee match (Delacretaz et al. 2016) • School Choice and bullying (Kasuya 2016) • Separating conflicting groups in refugee match • Daycare/nursery school matching: more on this later

Sufficiency for SOFM

• Similar to the existence of SOSM in standard case

• note a stable matching may not exist

• Computation is easy (c.f. proof)

Theorem: If each school’s constraint is a general upper bound, then there exists an SOFM.

Proof (1)• Given our characterization theorem, we study fixed points of T.

• Under general upper bound, use Tarski’s fixed point theorem (below)

• A set is called a lattice if for any pair of elements, their “join” (least upper bound) and “meet” (greatest lower bound) both exist.

• Example: “set of cutoff profiles”={1,…n+1}m with the product order.

• In particular, there is a “largest” and “smallest” elements

Tarski’s Theorem (special case): Let X be a finite lattice and f: X→X be weakly increasing, i.e., x≤x’ implies f(x)≤f(x’).

Then the set of the fixed points of f is a finite lattice. In particular, there are largest and smallest fixed points.

Proof (2)

• Back to proof: We’ll show T is weakly increasing. Suppose P ≤P’. 1. If Ps <P’s, then Ts(P)≤Ps+1≤P’s ≤Ts(P’). 2. Suppose Ps =P’s.

• Demand for s is (weakly) larger if students face higher cutoffs at all other schools, so Ds(P) is a subset of Ds(P’).

• So, Ts(P)=Ps+1 implies Ts(P’)=P’s+1, thus Ts(P)=Ts(P’). • So T(P)≤T(P’).

• Smallest fixed point induces SOFM. QED

Tarski’s Theorem (from last slide): Let X be a finite lattice and f: X→X be weakly increasing, i.e., x≤x’ implies f(x)≤f(x’).

Then the set of the fixed points of f is a finite lattice. In particular, there are largest and smallest fixed points.

Algorithm• Tarski’s theorem gives an intuitive (and polynomial-time) algorithm.

• Start with lowest possible cutoff profile, P (i.e., every student is above the cutoff at every school)

• Then P≤T(P)

• Apply T repeatedly and get: P≤T(P)≤T(T(P))≤T3(P)≤T4(P)≤…

• At some point it stops at some P*, and

• T(P*)=P*; so it induces a feasible, IR, and fair matching

• For any fixed point P, P*≤P; P* corresponds to SOFM

More general constraints?• The “general upper-bound” includes many practical cases,

but not all (e.g., minimum constraints)

• Does SOFM exist more generally?

• Answer: “no” in a specific sense

Theorem: Suppose the constraint of a school s is not a general upper bound. Then there exist student preferences and capacity constraints at other schools s.t., SOFM does not exist.

Proof (1)

• Suppose the constraint at s is not a general upper bound.

• Consider two cases:

Proof (2)

• Case 1 (“easy” case): Suppose the empty matching (i.e., no one is matched) is infeasible at s.

• Assume all students find s unacceptable.

• Clearly, there is no feasible and IR matching.

Proof (3)• Case 2 (“less easy” case): Suppose the empty matching is feasible

• Note there is some set I’ of students and its subset I’’ such that I’ is feasible but I’’ is not (and both are nonempty).

• Fix s’≠s and assume preferences

• students in I’’: s, s’

• students in I’\I’’: s’, s

• all other students find all schools unacceptable

• s’ has a large capacity

Proof (4)• Two fair (&feasible and IR) matchings:

1. everyone in I’ is matched to s and everyone else is unmatched

2. everyone in I’ is matched to s’ and everyone else is unmatched.

• If there is SOFM, then it should

• match everyone in I’’ to s, I’\I’’ to s’ and un-match everyone else

• Recall (from last slide)

• students in I’’: s, s’

• students in I’\I’’: s’, s

• all other students find all schools unacceptable

• s’ has a large capacity

→infeasible! QED

Application: Daycare Match • Some resources (teachers, rooms,

etc.) can be used for kids of different ages (Okumura 2017)

• Resource demand per kid varies across ages (younger kids need more teachers and space per capita)

→ general upper bound (while not capacity)

• Centralized matching algorithms.

• flexible assignments tried in several municipalities (but in ad hoc manners)

• Japan: daycare is greatly over-demanded

• Municipal governments are under great pressure to accommodate more children

Comparative statics

• Easy to prove, true more generally for arbitrary “relaxation of constraints”

• c.f. Results for SOSM in standard models (e.g., Crawford 1991; Konishi and Unver 2006)

• Flexibility across different ages will help.

• How about the magnitude?

Proposition: SOFM under flexible constraints is Pareto superior for students to SOFM under rigid constraints.

Daycare Match Simulation (in progress) • Data from Yamagata City (Yamagata)

and Bunkyo City (Tokyo), Japan:

• preferences (mechanism is strategy-proof)

• priorities

• outcomes

• We simulate SOFM under “flexible” and “rigid” constraints

Recall: SOFM under flexible constraints is Pareto superior to SOFM under rigid constraints.

Yamagata

Bunkyo

SOFM rigid SOFM flexible Actual outcome

Matched Children

Matched Unmatched(Data: Yamagata)

(1437 applicants in total)

SOFM rigid SOFM flexible Real allocation

SOFM rigid 0 14

SOFM flexible 869 238

Real allocation 659 72

(i,j) entry = #{applicants strictly better off in mechanism i than j}

(1437 applicants in total)

(Data: Yamagata)

SOFM rigid SOFM flexible Real allocation

pairs with envy 0 0 989

applicants with envy 0 0 475

daycare with envy 0 0 62

(1437 applicants, 93 daycares in total)

(Data: Yamagata)

Rank distribution

1 2 3 4 5 6 7 8

Rank Distribution

SOFM rigid SOFM flexible Actual outcome

1 2 3 4 5 6 7 8 9

SOSM flexible v. actual

Actual outcome SOFM flexible

(Data: Yamagata)

Extension: Tie in priority• College admission in Hungary (Biro 2010) uses a mechanism

like deferred acceptance, but

• Ranking over students are based on test score → ties

• Admitting all students with a score is infeasible → reject all students of that score

• Disaster shelter in Kobe and Tohoku earthquakes (Hayashi 2003, Hayashi 2011)

• Priorities include lots of ties (e.g., own house livable or not)

• Insufficient food supply was not allocated

Problems with ties:

• A has capacity of 1

• A ranks 1 and 2 equally

• But our theory extends: SOFM exists, etc.

• Characterization: fair and non-wastefulness are compatible iff capacity constraints and no ties.

A A 1,2

∅ ∅ ∅

Stability: Maximal domain

• Note: “necessary and sufficient” condition for stable matching existence

• The conclusion holds for any priorities and constraints at other schools. 84

Theorem: Suppose the constraint of a school s is not a capacity constraint (while being a general upper-bound). Then there exist a priority at s and student preferences s.t. there exists no stable matching.

• Recall stability (=Feasibility & IR & Fairness & Non-wastefulness) leads to non-existence. In fact,

Strategic issues

• But

• The same impossibility holds for any mechanism with feasibility, fairness, and unanimity.

• Approximate incentive compatibility holds in large markets. 85

Theorem: Suppose the constraint of a school s is not a capacity constraint. Then there are school priorities and standard capacity constraints at other schools such that the SOFM mechanism isn’t strategy-proof for students.

• SOFM mechanism isn’t necessarily strategy-proof for students

• Capacity constraints → SP for students

• Turns out this is “necessary” as well.

Related literature

• Distributional Constraints: Sonmez-Unver (2006), Biro-Fleiner-Irving-Manlove (2010 TCS), Kamada-Kojima (2015 AER, 2016 JET, 2017 TE), Milgrom (2009 AEJ Micro), Budish-Che-Kojima-Milgrom (2013 AER), Che-Kim-Mierendorff (2013 ECMA), Akbarpour-Nikzad (2016), Fragiadakis-Troyan (2016 TE), Goto-Hashimoto-Iwasaki-Kawasaki-Ueda-Yasuda-Yokoo (2014 AAMAS), Goto-Kojima-Kurata-Tamura-Yokoo (2016 AEJ), Kojima-Tamura-Yokoo (2018 JET),

• Affirmative action/diversity: Roth (1991 AER), Abdulkadiroglu-Sonmez (2003 AER), Abdulkadiroglu (2005 IJGT), Aygun-Turhan (2016), Dur-Pathank-Sonmez (2016), Ergin-Sonmez (2006 JPubE), Abdulkadiroglu-Pathak-Roth (2009 AER), Kojima (2012 GEB), Ehlers-Hafalir-Yenmez-Yildirim (2014 JET), Echenique-Yenmez (2015 AER), Hafalir-Yenmez-Yildirim (TE 2013), Westkamp (2010 ET), Sonmez (2013 JPE), Sonmez-Switzer (2013 ECMA), Dur-Kominers-Pathak-Sonmez (2016 JPE), Delacretaz-Kominers-Teytelboym (2016), Hassidim-Romm-Shorrer (2016), Milgrom-Segal (2016), Okumura (2017)

• Fairness: Foley (1967), Balinski-Sonmez (1999 JET), Sotomayor (1996 GEB), Blum-Roth-Rothblum (JET 1997), Wu-Roth (2017 GEB), Kesten-Yacizi (2010 ET), Biro (2010)

Conclusion• Characterization of fair matchings via a cutoff adjustment function

• The general upper-bound is the most general condition to guarantee existence of SOFM

• Daycare match application

• Future research

• Solution under non-general upper bounds

• More numerical and empirical study (new data just granted)

• Implementing the new design

What does “stability” mean?

What is the “right” stability concept?

Observation: standard stability may be incompatible with constraints

Weaken the concept; how?

General Model of constraints

Standard two-sided matching except

Set of “regions” (or constraint structures) R in 2H, assume each {h} is in R

Each region has exogenous regional cap

A matching is feasible if, for each region,

(# of doctors in the region) ≦ (regional cap)

(positive integer)

Strong stability

Definition: A matching is strongly stable (under constraints) if

Strong stability

Definition: A matching is strongly stable (under constraints) if 1. it is feasible,

Strong stability

Definition: A matching is strongly stable (under constraints) if 1. it is feasible,2. it is individually rational, and

Strong stability

Definition: A matching is strongly stable (under constraints) if 1. it is feasible,2. it is individually rational, and3. if (i,h) is a blocking pair,

Strong stability

Definition: A matching is strongly stable (under constraints) if 1. it is feasible,2. it is individually rational, and3. if (i,h) is a blocking pair,

then matching i and h is infeasible.

(and letting them reject current partners if they want)

Non-existenceLet there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

Each hospital has 1 seat

1 2 A B

A B 2 1

B A 1 2

1 2 A B

A B 2 1

B A 1 2

1 2 A B

A B 2 1

B A 1 2

�91❤

1 2 A B

A B 2 1

B A 1 2

1 2 A B

A B 2 1

B A 1 2

�91❤

1 2 A B

A B 2 1

B A 1 2

1 2 A B

A B 2 1

B A 1 2

�91❤

1 2 A B

A B 2 1

B A 1 2

AB 1 2

1 2 A B

A B 2 1

B A 1 2

AB 1 2

1 2 A B

A B 2 1

B A 1 2

1 2 A B

A B 2 1

B A 1 2

Non-existence!�91

Characterization for existence

Characterization for existenceWhen do the constraints guarantee existence of a strongly stable matching?

i.e., there exists a strongly stable matching for any set of doctors and (doctor and hospital) preferences.

trivial cases: for each region, either (1) the region has just one hospital or (2) its regional cap is zero.

Proposition (Kamada and Kojima, forthcoming, JET): Guaranteeing the existence of a strongly stable matching. ⬍Constraint structure is trivial.

Additional impossibility results

Another criterion

Require “the mechanism chooses a strongly stable matching whenever one exists”

Another criterion

By definition such a mechanism exists

Another criterion

By definition such a mechanism exists

Require also strategy-proofness for doctors (as in DA and FDA)

Strategy-proofnessLet there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

1 2 A B

A B 2 1

∅ ∅ 1 2

Not strategy-proof!�94

Characterization for mechanism

Proposition: There exists a mechanism that is strategy-proof for doctors and produces a strongly stable matching whenever one exists.　　　　　　　　　⬍Constraint structure is trivial.

Weakening stability

We want to weaken the stability concept

Weakening stability

We want the concept to have good properties:

Weakening stability

existence

Weakening stability

existence

compatible with strategy-proofness for doctors

Weakening stability

existence

strong enough to eliminate bad matchings

Weakening stability

existence

strong enough to eliminate bad matchings

imply efficiency

Regional preferences

Kamada and Kojima (2017, TE)

Assume information is available about regions’ rationing criteria:

Regional cap implies a region may need to ration matching of doctors across hospitals in it.

So assume, for each region and its largest partition by subregions, there is a regional preference, a weak ordering on number of doctors in those subregions.

Interpretation: policy goals (regarding rationing)

Assume regional preferences are substitutable and acceptant

So assume, for each region and its largest partition by subregions, there is a regional preference, a weak ordering on number of doctors in those subregions.

Interpretation: policy goals (regarding rationing)

Assume regional preferences are substitutable and acceptant

We say blocking (i,h) is illegitimate if a region where i and h reside does not prefer the move (and it involves h’s vacant position).

Illegitimate blocks: general case

Say that a blocking (i,h) is a Pareto improvement for a set R’ of regions if each region in R’ weakly prefers the new doctor distribution, with at least one strictly so.

Given a matching, blocking (i,h) is illegitimate if there is a region r such that:

the regional cap of r is currently full

blocking (i,h) is not a Pareto improvement for the set of all regions r’ ⊆ r such that both h and i are in r’

Stability

Definition: A matching is strongly stable if 1. it is feasible,2. it is individually rational, and3. if (i,h) is a blocking pair,

Stability

Definition: A matching is strongly stable if 1. it is feasible,2. it is individually rational, and3. if (i,h) is a blocking pair,

or illegitimate

Non-existence example revisitedLet there be one region, with cap 1.

Regional preference: (1,0)≻(0,1)

1 2 A B

A B 2 1

B A 1 2

�100

1 2 A B

A B 2 1

B A 1 2

�100

1 2 A B

A B 2 1

B A 1 2

�100

1 2 A B

A B 2 1

B A 1 2

stable!�100

Main Result

�101

Main ResultNecessary and sufficient condition in terms of the set of regions R

�101

Theorem: Given R, there exists a mechanism that is stable and strategy-proof for doctors (for all regional cap profiles and regional preferences)

⬍R is a hierarchy

�101

Theorem: Given R, there exists a mechanism that is stable and strategy-proof for doctors (for all regional cap profiles and regional preferences)

⬍R is a hierarchy

Corollary: If R is a partition, then a stable matching exists.

Proof idea: Hierarchy

�102

Proof idea: HierarchyUse a (generalized) “flexible deferred acceptance (FDA)” algorithm (generalizing Kamada and Kojima 2015)

�102

Associate a given problem of matching with constraints with matching with contracts (Hatfield and Milgrom 2005) between doctors and “the hospital consortium”

�102

Choice function satisfies substitutability (or substitutable completability) and law of aggregate demand (Hatfield and Kominers 2012, 2015)

�102

FDA corresponds to a (generalized) DA in matching with contracts

�102

FDA corresponds to a (generalized) DA in matching with contracts

Given hierarchy, verify sufficient conditions for stability + doctor-strategy-proofness in matching with contracts

�102

Proof idea: Non-hierarchyIllustrate the main idea by an example here

1 2 A B C

C B 2 1 2

A 1 2 1

�103

r r’ regional caps=1

r: (1,0)≻(0,1)

r’: (1,0)≻(0,1)

Non-hierarchy (cont.)

Two stable matchings:A B 2C

A 2 B C 1

Case 1:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

Two stable matchings:

Case 1:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

Case 1:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

oruniquely stable under misreport

Case 2:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

(→Back)

Case 2:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

(→Back)

Case 2:

A B 2C

A 2 B C 1

A B 2C

A 2 B C 1

uniquely stableunder misreport (→Back)

Stability is strong enough to reject some bad outcomes

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

�107

Stability is strong enough to reject some bad outcomes

1 2 3 4 … 8 9 10 A B

A A A B B B B 1 1∅ ∅ ∅ ∅ ∅ ∅ ∅ 2 2

⋮ ⋮

1 2 3 4 5 6 7 8

9 10stability violated!�107

Efficiency

�108

Proposition: A stable matching is (constrained) efficient.

Efficiency

�108

In usual stable matching problem, stability is equivalent to core, so efficiency follows as a corollary

Efficiency

�108

Not in this setting

Efficiency

�108

Not in this setting

No natural cooperative game; direct proof

General model of constraints

�109

Kamada and Kojima (forthcoming, JET)Goto, Kojima, Kurata, Tamura, and Yokoo (forthcoming, AEJ Micro)

More general model of constraints

�109

There is a “constraint function” f:Zn→{0,1}, (n=#{hospitals})

input= vector of #{doctors} placed across hospitals

1 and 0 mean “feasible” and “infeasible” respectively

f(w’)=1 & w≦w’ ➔ f(w)=1 (called “heredity” in discrete math)

�109

A “weakly stable matching” exists (Kamada and Kojima, forthcoming).

�109

A “weakly stable matching” exists (Kamada and Kojima, forthcoming).

Goto, Kojima, Kurata, Tamura, and Yokoo (forthcoming) find a mechanism that is strategy-proof for doctors.

�109

Matching and Discrete Convex Analysis

�110

(Kojima, Tamura, and Yokoo 2016)

There are many types of constraints in practice

Regional cap = maximum quotas on sets of hospitals

Minimum quotas (on individuals or sets of hospitals)

“Affirmative action” constraints

�110

There are many types of constraints in practice

Regional cap = maximum quotas on sets of hospitals

Minimum quotas (on individuals or sets of hospitals)

“Affirmative action” constraints

Use discrete convex analysis (a branch of discrete math) to obtain unified results and new applications.

Key: “M♮-concavity” (concavity for discrete domains) of hospital preference

�110

Related literatureDistributional Constraints: Biro-Fleiner-Irving-Manlove (2010 TCS), Milgrom (2009 AEJ Micro), Budish-Che-Kojima-Milgrom (2013 AER), Che-Kim-Mieremdorff (2013 ECMA), Akbarpour-Nikzad (2016), Fragiadakis-Troyan (2016 TE), Goto-Hashimoto-Iwasaki-Kawasaki-Ueda-Yasuda-Yokoo (2014 AAMAS).

Affirmative action/diversity: Roth (1991 AER), Abdulkadiroglu-Sonmez (2003 AER), Abdulkadiroglu (2005 IJGT), Aygun-Turhan (2016), Dur-Pathank-Sonmez (2016 mimeo), Ergin-Sonmez (2006 JPubE), Abdulkadiroglu-Pathak-Sonmez (2009 AER), Kojima (2012 GEB), Ehlers-Hafalir-Yenmez-Yildirim (2014 JET), Echenique-Yenmez (2015 AER), Hafalir-Yenmez-Yildirim (TE 2013), Westkamp (2010 ET), Sonmez (2013 JPE), Sonmez-Switzer (2013 ECMA), Dur-Kominers-Pathak-Sonmez (2014), Dur-Pathak-Sonmez (2016), Delacretaz-Kominers-Teytelboym (2016), Hassidim-Romm-Shorrer (2016)

Matching with contracts: Kelso-Crawford (1982 ECMA), Fleiner (2003 MOR), Hatfield-Milgrom (2005 AER), Echenique (2012 AER), Hatfield-Kojima (2008 AER, 2009 GEB, 2010 JET), Ostrovsky (2008 AER), Hatfield-Kominers-Nichifor-Ostrovsky-Westkamp (2013 JPE, 2015a,b mimeo), Hatfield-Kominers (2015 mimeo), Hatfield-Kominers-Westkamp (2016)

�111

Conclusion

�112

Conclusion

�112

Conclusion

�112

Conclusion

�112

Conclusion

�112

Conclusion

Interaction with markets in practice continues to enrich theory.

�112

Additional Slides

How robust is approximate IC?

�114

Rely on various assumptions

�114

I focus on “short list”, k=constant

�114

Motivated by real-life markets

�114

Motivated by real-life markets

k<15 (NRMP), 12 (NYC), 8 (APPIC), 4 (JRMP), …

�114

“Necessity” of short lists

Simulation for k=n

�115

Knuth, Motwani, and Pittel (1990 RSA)

Simulation for k=n

�115

Intuition: Rejection chain doesn’t get absorbed by vacancy!

Simulation for k=n

�115

Intuition: Rejection chain doesn’t get absorbed by vacancy!

Then, is approximate IC sensitive to the short list assumption?Simulation for k=n

�115

Approximate IC with long lists

�116Back to Setup, Intuition

Lee (2014): Many agents have profitable manipulations, but utility gains are small

Consistent with previous results

Ashlagi, Kanoria, and Leshno (2015 JPE): Assume the market is unbalanced, i.e. #{doctors}≠#{hospitals}

With vacancy, rejection chain works (and much better than previously shown!)

Failure of approximate IC is a “knife edge” case

Preference restriction

�117

Klaus and Klijn (2005 JET) find a sufficient and “almost necessary” condition for couple preferences: responsiveness

�117

Klaus and Klijn (2005 JET) find a sufficient and “almost necessary” condition for couple preferences: responsiveness

APPIC data: 1/167 couples have responsive preferences (Kojima, Pathak, and Roth 2013 QJE)

�117

Explaining data better

�118Back

Kojima et al. (2013) result obtained with assumption #{couples}=o(√n), so couples quickly vanishes in proportion.

�118Back

Ashlagi, Braverman, and Hassidim (2014 OR) improved the result to #{couples}=o(n).

�118Back

Also nonexistence probability is bounded below if couples proportion is constant.

�118Back

Also nonexistence probability is bounded below if couples proportion is constant.

Nguyen and Vohra (2016): Scarf’s Lemma, rounding.

�118Back

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