Matching with Constraints

Matching with Constraints

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

“General theory of matching with constraints” mimeo, 2015b

“Stability concepts in matching under distributional constraints” mimeo, 2015c

based on joint works with Yuichiro Kamada,

1

Overview

Many matching markets are subject to constraints

Medical specialties

Multiple school programs sharing one building

Affirmative action (diversity constraints)

Specific real-life examples

Japanese residency match: regional cap (→details)

Other examples (→details)

Goal of this talk

Goal of this talk

Main questions:

Goal of this talk

Main questions:

Desirable design under constraints?

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

a new mechanism solves the problems

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

a new mechanism solves the problems

Papers 2&3: We study stability in more detail

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

a new mechanism solves the problems

Papers 2&3: We study stability in more detail

a seemingly natural definition doesn’t work

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

a new mechanism solves the problems

Papers 2&3: We study stability in more detail

a seemingly natural definition doesn’t work

“stability” is defined using the idea of rationing

Goal of this talk

Main questions:

Desirable design under constraints?

What properties are theoretically possible?

Paper 1: We find existing mechanisms in practice suffer from inefficiency and instability

a new mechanism solves the problems

Papers 2&3: We study stability in more detail

a seemingly natural definition doesn’t work

“stability” is defined using the idea of rationing

Main result: Characterize the kind of constraints that allow for stable and doctor-strategy-proof mechanisms

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

4

Standard model: stability

Standard model: stability

A matching is stable if Individual rationality

No blocking pairs

Standard model: stability

A matching is stable if Individual rationality

No blocking pairs

Equivalent to core, so implies Pareto efficiency

Standard model: stability

A matching is stable if Individual rationality

No blocking pairs

Equivalent to core, so implies Pareto efficiency (doctor-proposing) deferred acceptance (DA):

Finds a stable matching

Strategy-proof for doctors

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)

6

Example: JRMP mechanism

Example: JRMP mechanism

Government imposes a target capacity for each hospital

Smaller than real (physical) hospital capacity

Sums at most to the regional caps

Example: JRMP mechanism

Government imposes a target capacity for each hospital

Smaller than real (physical) hospital capacity

Sums at most to the regional caps

JRMP mechanism = DA using target capacity

Feasible

7

Example: JRMP mechanism

Government imposes a target capacity for each hospital

Smaller than real (physical) hospital capacity

Sums at most to the regional caps

JRMP mechanism = DA using target capacity

Feasible

Is JRMP stable? (constrained) efficient?

7

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

⋮ ⋮

10 10

Each hospital

has 10 seats

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

Each hospital

has 10 seats

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

12 3 4 5 6 7 8

Each hospital

has 10 seats

9 10

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 5 6 7 8

Each hospital

has 10 seats

9 10

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 56 7 8

Each hospital

has 10 seats

9 10

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 56 7 8

Each hospital

has 10 seats

9 10

💔

💔

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 56 7 8

Each hospital

has 10 seats

9 10

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 56 7 8

Each hospital

has 10 seats

9 10

inefficient & unstable!

8

FDA mechanism

(flexible DA)

FDA mechanism

Begin with an empty matching, and repeat Steps below:

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

Acceptance/Rejection: Each hospital (tentatively) accepts from both its tentatively matched doctors and new applicants (if

any):

Phase 1 (“regular” phase): each hospital (tentatively) accepts its favorite acceptable applicants up to its target capacity.

(flexible DA)

FDA mechanism

Begin with an empty matching, and repeat Steps below:

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

Acceptance/Rejection: Each hospital (tentatively) accepts from both its tentatively matched doctors and new applicants (if

any):

Phase 1 (“regular” phase): each hospital (tentatively) accepts its favorite acceptable applicants up to its target capacity.

Phase 2 (“waitlist” phase): hospitals take turns to

(tentatively) accept favorite applicants from waitlist until (i) the regional cap becomes full or (ii) no doctor remains to be processed.

(flexible DA)

FDA example

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

⋮ ⋮

10 10

Each hospital

has 10 seats

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

Each hospital

has 10 seats

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

1 2 3 4 5 6 7 8 9 10

Each hospital

has 10 seats

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

1 2 3

4 5 6 7 8 9 10

Each hospital

has 10 seats

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

4 5 6 7 8 9 10

1 2 3

Each hospital

has 10 seats

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

4 5 6 7 8 9 10

1 2 3

Each hospital

has 10 seats

❤❤❤❤❤

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

4 5 6 7 8 9 10

1 2 3

Each hospital

has 10 seats

💛💛

❤❤❤❤❤

10 regular phase

FDA example

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

⋮ ⋮

10 10

FDA (target capacity=5 each):

A

B

4 5 6 7 8 9 10

1 2 3

Each hospital

has 10 seats

💛💛

efficient & stable!

❤❤❤❤❤

10 regular phase

FDA theoretical results

FDA theoretical results

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

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

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

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

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

11

What does “stability” mean?

What is the “right” stability concept?

Observation: standard stability may be incompatible with constraints

Weaken the concept; how?

12

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)

13

Strong stability

Strong stability

Definition:

A matching is

strongly stable

if

Strong stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

Strong stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

2. it is individually rational, and

Strong stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

2. it is individually rational, and

3. if (i,A) is a blocking pair,

Strong stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

2. it is individually rational, and

3. if (i,A) is a blocking pair,

then

matching i and A

is

infeasible

.

14

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

₂

1

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

1

₂

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

1

₂

Each hospital

has 1 seat

15

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

2

1

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

2

1

Each hospital

has 1 seat

15

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

₂

1

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

₂

1

Each hospital

has 1 seat

15

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B 1

₂

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B 1

₂

Each hospital

has 1 seat

15

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

1

₂

Each hospital

has 1 seat

Non-existence

Let there be one region, with cap 1.

1 2 A B

A B 2 1

B A 1 2

A

B

1

₂

Each hospital

has 1 seat

Non-existence!

Characterization for existence

Characterization for existence

Say that the constraint structure (the family of regions)

Characterization for existence

Say that the constraint structure (the family of regions)

guarantees existence of a strongly stable matching if there exists a strongly stable matching for every

hospital capacities, number of doctors, and preferences.

Characterization for existence

Say that the constraint structure (the family of regions)

guarantees existence of a strongly stable matching if there exists a strongly stable matching for every

hospital capacities, number of doctors, and preferences.

independent across hospitals if for each region, either (1) the region has just one hospital or (2) its regional cap is zero. (i.e., only trivial constraints)

Characterization for existence

Say that the constraint structure (the family of regions)

guarantees existence of a strongly stable matching if there exists a strongly stable matching for every

hospital capacities, number of doctors, and preferences.

independent across hospitals if for each region, either (1) the region has just one hospital or (2) its regional cap is zero. (i.e., only trivial constraints)

16

Proposition:

Constraint structure guarantees

the existence of a strongly stable matching

⬍

Another criterion

Another criterion

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

Another criterion

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

By definition such a mechanism exists

Another criterion

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

By definition such a mechanism exists

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

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

₂

1

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

1

₂

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

1

₂

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

1

₂

Each hospital

has 1 seat

18 A

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

1

₂

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

₂

1

Each hospital

has 1 seat

Strategy-proofness

Let there be one region, with cap 1.

1 2 A B

A B 2 1

∅ ∅ 1 2

A

B

₂

1

Each hospital

has 1 seat

Not strategy-proof!

Characterization for mechanism

Recall the constraint structure is independent across hospitals if for each region, either (1) the region has just one hospital or (2) its regional cap is zero. (i.e., only trivial constraints)

Characterization for mechanism

Recall the constraint structure is independent across hospitals if for each region, either (1) the region has just one hospital or (2) its regional cap is zero. (i.e., only trivial constraints)

19

Proposition:

There exists a mechanism that is

strategy-proof for doctors and produces a strongly

stable matching whenever one exists.

_⬍

Weakening stability

Weakening stability

We want to weaken the stability concept

Weakening stability

We want to weaken the stability concept

We want the concept to have good properties:

Weakening stability

We want to weaken the stability concept

We want the concept to have good properties:

existence

Weakening stability

We want to weaken the stability concept

We want the concept to have good properties:

existence

compatible with strategy-proofness for doctors

Weakening stability

We want to weaken the stability concept

We want the concept to have good properties:

existence

compatible with strategy-proofness for doctors

strong enough to eliminate bad matchings

Weakening stability

We want to weaken the stability concept

We want the concept to have good properties:

existence

compatible with strategy-proofness for doctors

strong enough to eliminate bad matchings

imply efficiency

Regional preferences

A region may need to ration matching of doctors across hospitals in it.

So we assume, for each region r and its largest partition by subregions S, there is a regional preference ≽_r,S, a

weak ordering on # of doctors at immediate successors.

Interpretation: policy goals (regarding rationing)

Assume regional preferences are substitutable and acceptant

Illegitimate blocks

How to reflect regional preferences into stability?

Say that a blocking (i,A) 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,A) is illegitimate if there is a region r such that:

the regional cap of r is currently full

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

Stability

Stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

2. it is individually rational, and

3. if (i,A) is a blocking pair,

then

matching i and A

is

infeasible

.

Stability

Definition:

A matching is

strongly stable

if

1. it is feasible,

2. it is individually rational, and

3. if (i,A) is a blocking pair,

then

matching i and A

is

infeasible

.

23

Non-existence example revisited

Let there be one region, with cap 1.

Regional preference: prefer (1,0) to (0,1)

1 2 A B

A B 2 1

B A 1 2

Each hospital

has 1 seat

Non-existence example revisited

Let there be one region, with cap 1.

Regional preference: prefer (1,0) to (0,1)

1 2 A B

A B 2 1

B A 1 2

A

B

₂

1

Each hospital

has 1 seat

Non-existence example revisited

Let there be one region, with cap 1.

Regional preference: prefer (1,0) to (0,1)

1 2 A B

A B 2 1

B A 1 2

A

B

2

1

Each hospital

has 1 seat

Non-existence example revisited

Let there be one region, with cap 1.

Regional preference: prefer (1,0) to (0,1)

1 2 A B

A B 2 1

B A 1 2

A

B

2

1

Each hospital

has 1 seat

stable!

Characterization Result

Characterization Result

Characterization is in terms of regions R

Characterization Result

Characterization is in terms of regions R

25

Theorem:

For all regional cap profiles and

regional preferences given R, there exists a

mechanism that is stable and strategy-proof

for doctors

⬍

Characterization Result

Characterization is in terms of regions R

25

Theorem:

For all regional cap profiles and

regional preferences given R, there exists a

mechanism that is stable and strategy-proof

for doctors

⬍

R is a hierarchy

Proof idea: Hierarchy

Proof idea: Hierarchy

Generalize FDA

Proof idea: Hierarchy

Generalize FDA

Associate a given problem of matching with

constraints with matching with contracts (Hatfield and Milgrom 2005) between doctors and regions

Proof idea: Hierarchy

Generalize FDA

Associate a given problem of matching with

constraints with matching with contracts (Hatfield and Milgrom 2005) between doctors and regions

FDA corresponds to a (generalized) DA in matching with contracts (recall simple case before)

Proof idea: Hierarchy

Generalize FDA

Associate a given problem of matching with

constraints with matching with contracts (Hatfield and Milgrom 2005) between doctors and regions

FDA corresponds to a (generalized) DA in matching with contracts (recall simple case before)

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

Proof idea: Non-hierarchy

Illustrate the main idea by an example here

1 2 A B C

C B 2 1 2

A 1 2 1

Each hospital

has 1 seat

27

A B C

r

r’

regional caps=1

(1,0)≻r(0,1)

Non-hierarchy (cont.)

Two stable matchings:

A

B 2

C

A 2

Non-hierarchy (cont.)

Case 1:

A

B 2

C

A 2

B

C 1

A

B 2

C

A 2

B

C 1

or

Non-hierarchy (cont.)

Case 1:

A

B 2

C

A 2

B

C 1

A

B

A

B 2

C

A 2

B

C 1

or

Non-hierarchy (cont.)

Case 1:

A

B 2

C

A 2

B

C 1

A

B

A

B 2

C

A 2

B

C 1

Non-hierarchy (cont.)

Case 2:

A

B 2

C

A 2

B

C 1

A

B 2

C

A 2

B

C 1

or

Non-hierarchy (cont.)

Case 2:

C

A

A

B 2

C

A 2

B

C 1

A

B 2

C

A 2

B

C 1

or

Non-hierarchy (cont.)

Case 2:

C

A

A

B 2

C

A 2

B

C 1

A

B 2

C

A 2

B

C 1

or

Two stable matchings:

uniquely stable

Stability is strong enough to

reject some bad outcomes

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

⋮ ⋮

10 10

Each hospital

has 10 seats

Stability is strong enough to

reject some bad outcomes

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

⋮ ⋮

10 10

JRMP (target capacity=5 each):

A

B

1 2 3

4 5 6 7 8

Each hospital

has 10 seats

9 10

stability violated!

31

Efficiency

32

Efficiency

32

Proposition:

A stable matching is

(constrained) efficient.

Efficiency

32

Proposition:

A stable matching is

(constrained) efficient.

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

Efficiency

32

Proposition:

A stable matching is

(constrained) efficient.

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

Not in this setting

More general model of

constraints

More general model of

constraints

More general model of constraints

More general model of

constraints

More general model of constraints

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; “heredity” in discrete math

More general model of

constraints

More general model of constraints

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; “heredity” in discrete math

Our result says that there is no mechanism and doctor-strategy-proof mechanism in general

More general model of

constraints

More general model of constraints

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; “heredity” in discrete math

Our result says that there is no mechanism and doctor-strategy-proof mechanism in general

A “weakly stable” matching exists (Kamada and Kojima, 2015c)

See also Goto, Kojima, Kurata, Tamura, and Yokoo (in progress)

Matching and Discrete Convex Analysis

34

Matching and Discrete Convex Analysis

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

34

Matching and Discrete Convex Analysis

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

34

Related literature

Distributional Constraints: Biro-Fleiner-Irving-Manlove (2010 TCS),

Budish-Che-Kojima-Milgrom (2013 AER), Fragiadakis-Troyan (2014 mimeo), Goto-Hashimoto-Iwasaki-Kawasaki-Ueda-Yasuda-Yokoo (2014 AAMAS)

Affirmative action: Roth (1991 AER), Abdulkadiroglu-Sonmez (2003 AER), Abdulkadiroglu (2005 IJGT), 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)

Matching with contracts: Kelso-Crawford (1982 ECMA), Fleiner (2003 MOR), Milgrom (2005 AER), Echenique (2012 AER),

Kojima (2008 AER, 2009 GEB, 2010 JET), Ostrovsky (2008 AER), Hatfield-Kominers-Nichifor-Ostrovsky-Westkamp (2013 JPE, 2015a,b mimeo),

Hatfield-Kominers (2015 mimeo)

Conclusion

Many markets are subject to constraints

Classic theory doesn’t directly apply to practical markets; motivate new theory

Existing mechanisms don’t work well

New mechanism: FDA

“Strong stability” leads to impossibilities

Rationing criterion (policy goals) lead do “stability”

Stable and strategy-proof mechanism possible iff regions form a hierarchy

Future research: Design for non-hierarchy cases?

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Case study: Japan

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Case study: Japan

Japan residency matching program (JRMP)

adopted doctor-proposing deferred acceptance mechanism (DA) in 2003

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Case study: Japan

Japan residency matching program (JRMP)

adopted doctor-proposing deferred acceptance mechanism (DA) in 2003

Critics claimed rural hospitals cannot fill enough positions under DA.

38

Case study: Japan

Japan residency matching program (JRMP)

adopted doctor-proposing deferred acceptance mechanism (DA) in 2003

Critics claimed rural hospitals cannot fill enough positions under DA.

Government introduced a regional cap as a constraint (→numbers)

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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)

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Japanese Data on Regional Caps

*** paste the prefecture picture ***

!500$ 0$ 500$ 1000$ 1500$ 2000$ Tok yo$ O sak a$ Kan ag aw a$ Ai ch i$ Fu ku ok a$ Ho kk ai do $ Hy og o$ Ch ib a$ Ky oto $ Sai tam a$ Sh izu ok a$ Hi ro sh im a$ O kay am a$ N ag an o$ Mi yag i$ Ib ar ak i$ O ki naw a$ Tochig i$ Gi fu $ G un m a$ N iig

ata$ _Mie$

N ag as ak i$ Ku m am oto $ Fu ku sh im a$ Kag os hi m a$ Is hi kaw a$ Yam ag uc hi $ Ak ita$ N ar a$ Ehim e$ To yam a$ Ao mo ri$ Iw ate $ Yam ag

ata$ _Oita$

Simulation:

number of matched/unmatched doctors

0" 1000" 2000" 3000" 4000" 5000" 6000" 7000" 8000" 9000"

DA" JRMP" FDA"

Num be r'of 'doc tor s' Mechanism' Unmatched"doctors" Matched"doctors" 41 Matched Unmatched unconstrained

DA JRMP FDA

Simulation: Rank distributions

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

Num be r'of 'doc tor s' Ranking'of'the'matched'hospital' DA# FDA# JRMP# 42 DA FDA JRMP N um be r of do ct or s

Ranking of matched hospitals