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Recent Advances in Generalized Matching Theory John William Hatfield Stanford Graduate School of Business Scott Duke Kominers Becker Friedman Institute, University of Chicago Matching Problems: Economics meets Mathematics Conference

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Recent Advances in Generalized Matching Theory

John William Hatfield

Stanford Graduate School of Business

Scott Duke Kominers

Becker Friedman Institute, University of Chicago

“Matching Problems: Economics meets Mathematics” Conference

Becker Friedman Institute & Stevanovich Center, University of Chicago

June 4, 2012

Hatfield & Kominers June 4, 2012 1

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Generalized Matching Theory Stable Marriage

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with a set of men M and a set of women W , how can we arrange marriages so that no agent wishes for a divorce?

Hatfield & Kominers June 4, 2012 2

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Generalized Matching Theory Stable Marriage

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with a set of men M and a set of women W , how can we arrange marriages so that no agent wishes for a divorce?

Assumptions

1 Bilateral relationships: only pairs (and possibly singles). 2 Two-sided: men only desire women; women only desire men. 3 Preferences are fully known. Hatfield & Kominers June 4, 2012 2

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Generalized Matching Theory Stable Marriage

The Deferred Acceptance Algorithm

Step 1

1 Each man “proposes” to his first-choice woman. 2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

Hatfield & Kominers June 4, 2012 3

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Generalized Matching Theory Stable Marriage

The Deferred Acceptance Algorithm

Step 1

1 Each man “proposes” to his first-choice woman. 2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

Step t ≥ 2

1 Each rejected man “proposes” to the his favorite woman who

has not rejected him.

2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

Hatfield & Kominers June 4, 2012 3

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Generalized Matching Theory Stable Marriage

The Deferred Acceptance Algorithm

Step 1

1 Each man “proposes” to his first-choice woman. 2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

Step t ≥ 2

1 Each rejected man “proposes” to the his favorite woman who

has not rejected him.

2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

At termination, no agent wants a divorce!

Hatfield & Kominers June 4, 2012 3

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Generalized Matching Theory Stable Marriage

Stability

Definition

A matching µ is a one-to-one correspondence on M ∪ W such that µ(m) ∈ W ∪ {m} for each m ∈ M, µ(w) ∈ M ∪ {w} for each w ∈ W , and µ2(i) = i for all i ∈ M ∪ W .

Definition

A marriage matching µ is stable if no agent wants a divorce.

Hatfield & Kominers June 4, 2012 4

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Generalized Matching Theory Stable Marriage

Stability

Definition

A matching µ is a one-to-one correspondence on M ∪ W such that µ(m) ∈ W ∪ {m} for each m ∈ M, µ(w) ∈ M ∪ {w} for each w ∈ W , and µ2(i) = i for all i ∈ M ∪ W .

Definition

A marriage matching µ is stable if no agent wants a divorce: Individually Rational: All agents i find their matches µ(i) acceptable. Unblocked: There do not exist m, w such that both m ≻w µ(w) and w ≻m µ(m).

Hatfield & Kominers June 4, 2012 4

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Generalized Matching Theory Stable Marriage

Existence and Lattice Structure

Theorem (Gale–Shapley, 1962)

A stable marriage matching exists.

Hatfield & Kominers June 4, 2012 5

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Generalized Matching Theory Stable Marriage

Existence and Lattice Structure

Theorem (Gale–Shapley, 1962)

A stable marriage matching exists.

Theorem (Conway, 1976)

Given two stable matchings µ, ν, there is a stable match µ ∨ ν (µ ∧ ν) which every man likes weakly more (less) than µ and ν.

Hatfield & Kominers June 4, 2012 5

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Generalized Matching Theory Stable Marriage

Existence and Lattice Structure

Theorem (Gale–Shapley, 1962)

A stable marriage matching exists.

Theorem (Conway, 1976)

Given two stable matchings µ, ν, there is a stable match µ ∨ ν (µ ∧ ν) which every man likes weakly more (less) than µ and ν. If all men (weakly) prefer stable match µ to stable match ν, then all women (weakly) prefer ν to µ.

Hatfield & Kominers June 4, 2012 5

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Generalized Matching Theory Stable Marriage

Existence and Lattice Structure

Theorem (Gale–Shapley, 1962)

A stable marriage matching exists.

Theorem (Conway, 1976)

Given two stable matchings µ, ν, there is a stable match µ ∨ ν (µ ∧ ν) which every man likes weakly more (less) than µ and ν. If all men (weakly) prefer stable match µ to stable match ν, then all women (weakly) prefer ν to µ. The man- and woman-proposing deferred acceptance algorithms respectively find the man- and woman-optimal stable matches.

Hatfield & Kominers June 4, 2012 5

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Generalized Matching Theory Stable Marriage

Opposition of Interests: A Simple Example

≻m1 : w1 ≻ w2 ≻ ∅ ≻m2 : w2 ≻ w1 ≻ ∅ ≻w1 : m2 ≻ m1 ≻ ∅ ≻w2 : m1 ≻ m2 ≻ ∅

Hatfield & Kominers June 4, 2012 6

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Generalized Matching Theory Stable Marriage

Opposition of Interests: A Simple Example

≻m1 : w1 ≻ w2 ≻ ∅ ≻m2 : w2 ≻ w1 ≻ ∅ ≻w1 : m2 ≻ m1 ≻ ∅ ≻w2 : m1 ≻ m2 ≻ ∅ man-optimal stable match

Hatfield & Kominers June 4, 2012 6

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Generalized Matching Theory Stable Marriage

Opposition of Interests: A Simple Example

≻m1 : w1 ≻ w2 ≻ ∅ ≻m2 : w2 ≻ w1 ≻ ∅ ≻w1 : m2 ≻ m1 ≻ ∅ ≻w2 : m1 ≻ m2 ≻ ∅ man-optimal stable match woman-optimal stable match

Hatfield & Kominers June 4, 2012 6

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Generalized Matching Theory Stable Marriage

Opposition of Interests: A Simple Example

≻m1 : w1 ≻ w2 ≻ ∅ ≻m2 : w2 ≻ w1 ≻ ∅ ≻w1 : m2 ≻ m1 ≻ ∅ ≻w2 : m1 ≻ m2 ≻ ∅ man-optimal stable match woman-optimal stable match This opposition of interests result also implies that there is no mechanism which is strategy-proof for both men and women.

Hatfield & Kominers June 4, 2012 6

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Generalized Matching Theory Stable Marriage

The “Lone Wolf” Theorem (McVitie–Wilson, 1970)

Theorem

The set of matched men (women) is invariant across stable matches.

Proof

Hatfield & Kominers June 4, 2012 7

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Generalized Matching Theory Stable Marriage

The “Lone Wolf” Theorem (McVitie–Wilson, 1970)

Theorem

The set of matched men (women) is invariant across stable matches.

Proof

Let ¯ µ be the man-optimal stable match and µ be any stable match.

Hatfield & Kominers June 4, 2012 7

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Generalized Matching Theory Stable Marriage

The “Lone Wolf” Theorem (McVitie–Wilson, 1970)

Theorem

The set of matched men (women) is invariant across stable matches.

Proof

Let ¯ µ be the man-optimal stable match and µ be any stable match. Every man weakly prefers ¯ µ; the number of married men under ¯ µ is weakly greater than under µ.

Hatfield & Kominers June 4, 2012 7

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Generalized Matching Theory Stable Marriage

The “Lone Wolf” Theorem (McVitie–Wilson, 1970)

Theorem

The set of matched men (women) is invariant across stable matches.

Proof

Let ¯ µ be the man-optimal stable match and µ be any stable match. Every man weakly prefers ¯ µ; the number of married men under ¯ µ is weakly greater than under µ. Every woman weakly prefers µ; the number of married women under µ is weakly greater than under ¯ µ.

Hatfield & Kominers June 4, 2012 7

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Generalized Matching Theory Stable Marriage

Applications

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Stable Marriage

Applications

National Residency Matching Program

“Men” are the medical students and “women” are the hospitals.

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Stable Marriage

Applications

National Residency Matching Program

“Men” are the medical students and “women” are the hospitals.

School choice

“Men” are the students and “women” are the schools.

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Stable Marriage

Applications

National Residency Matching Program

“Men” are the medical students and “women” are the hospitals.

School choice

“Men” are the students and “women” are the schools.

Labor markets

“Men” are the workers and “women” are the firms.

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Stable Marriage

Applications

National Residency Matching Program

“Men” are the medical students and “women” are the hospitals.

School choice

“Men” are the students and “women” are the schools.

Labor markets

“Men” are the workers and “women” are the firms.

Auctions

“Men” are the bidders and the “woman” is the auctioneer.

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Stable Marriage

Applications

National Residency Matching Program

“Men” are the medical students and “women” are the hospitals.

School choice

“Men” are the students and “women” are the schools.

Labor markets

“Men” are the workers and “women” are the firms.

Auctions

“Men” are the bidders and the “woman” is the auctioneer.

But in general these applications require that women take on multiple partners and that relationships take on many forms.

Hatfield & Kominers June 4, 2012 8

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

A set of hospitals H: each hospital has a strict preferences over subsets of contracts involving it, and

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

A set of hospitals H: each hospital has a strict preferences over subsets of contracts involving it, and A set of contracts X ⊆ D × H × T, where T is a finite set of terms such as wages, hours, etc.

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

A set of hospitals H: each hospital has a strict preferences over subsets of contracts involving it, and A set of contracts X ⊆ D × H × T, where T is a finite set of terms such as wages, hours, etc.

xD identifies the doctor of contract x;

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

A set of hospitals H: each hospital has a strict preferences over subsets of contracts involving it, and A set of contracts X ⊆ D × H × T, where T is a finite set of terms such as wages, hours, etc.

xD identifies the doctor of contract x; xH identifies the hospital of contract x.

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts

A set of doctors D: each doctor has a strict preference order

  • ver contracts involving him,

A set of hospitals H: each hospital has a strict preferences over subsets of contracts involving it, and A set of contracts X ⊆ D × H × T, where T is a finite set of terms such as wages, hours, etc.

xD identifies the doctor of contract x; xH identifies the hospital of contract x.

An outcome is a set of contracts Y ⊆ X such that if x, z ∈ Y and xD = zD, then x = z.

Hatfield & Kominers June 4, 2012 9

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Generalized Matching Theory Matching with Contracts

Choice Functions

Hatfield & Kominers June 4, 2012 10

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Generalized Matching Theory Matching with Contracts

Choice Functions

C d(Y ) ≡ maxPd{x ∈ Y : xD = d}.

Hatfield & Kominers June 4, 2012 10

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Generalized Matching Theory Matching with Contracts

Choice Functions

C d(Y ) ≡ maxPd{x ∈ Y : xD = d}. C h(Y ) ≡ maxPh{Z ⊆ Y : ZH = {h}}.

Hatfield & Kominers June 4, 2012 10

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Generalized Matching Theory Matching with Contracts

Choice Functions

C d(Y ) ≡ maxPd{x ∈ Y : xD = d}. C h(Y ) ≡ maxPh{Z ⊆ Y : ZH = {h}}. We define the rejection functions RD(Y ) ≡ Y − ∪d∈DC d(Y ), RH(Y ) ≡ Y − ∪h∈HC h(Y ).

Hatfield & Kominers June 4, 2012 10

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Generalized Matching Theory Matching with Contracts

Choice Functions

C d(Y ) ≡ maxPd{x ∈ Y : xD = d}. C h(Y ) ≡ maxPh{Z ⊆ Y : ZH = {h}}. We define the rejection functions RD(Y ) ≡ Y − ∪d∈DC d(Y ), RH(Y ) ≡ Y − ∪h∈HC h(Y ).

Definition

The preferences of hospital h are substitutable if for all Y ⊆ X, if z / ∈ C h(Y ), then z / ∈ C h({x} ∪ Y ) for all x = z.

Hatfield & Kominers June 4, 2012 10

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Generalized Matching Theory Matching with Contracts

Equilibrium

Definition

An outcome A is stable if it is

1 Individually rational:

for all d ∈ D, if x ∈ A and xD = d, then x ≻d ∅, for all h ∈ H, C h(A) = AH.

2 Unblocked: There does not exist a nonempty blocking set

Z ⊆ X − A and hospital h such that Z ⊆ C h (A ∪ Z) and Z ⊆ C D(A ∪ Z).

Hatfield & Kominers June 4, 2012 11

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Generalized Matching Theory Matching with Contracts

Equilibrium

Definition

An outcome A is stable if it is

1 Individually rational:

for all d ∈ D, if x ∈ A and xD = d, then x ≻d ∅, for all h ∈ H, C h(A) = AH.

2 Unblocked: There does not exist a nonempty blocking set

Z ⊆ X − A and hospital h such that Z ⊆ C h (A ∪ Z) and Z ⊆ C D(A ∪ Z). Stability is a price-theoretic notion:

Every contract not taken . . . . . . is available to some agent who does not choose it.

Hatfield & Kominers June 4, 2012 11

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Generalized Matching Theory Matching with Contracts

Characterization of Stable Outcomes

Consider the operator ΦH

  • X D

≡ X − RD

  • X D

ΦD

  • X H

≡ X − RH

  • X H

Φ

  • X D, X H

=

  • ΦD
  • X H

, ΦH

  • X D

Hatfield & Kominers June 4, 2012 12

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Generalized Matching Theory Matching with Contracts

Characterization of Stable Outcomes

Consider the operator ΦH

  • X D

≡ X − RD

  • X D

ΦD

  • X H

≡ X − RH

  • X H

Φ

  • X D, X H

=

  • ΦD
  • X H

, ΦH

  • X D

Theorem

Suppose that the preferences of hospitals are substitutable. Then if Φ

  • X D, X H

=

  • X D, X H

, the outcome X D ∩ X H is stable. Conversely, if A is a stable outcome, there exist X D, X H ⊆ X such that Φ

  • X D, X H

=

  • X D, X H

and X D ∩ X H = A.

Hatfield & Kominers June 4, 2012 12

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Generalized Matching Theory Matching with Contracts

Existence of Stable Allocations

Theorem

Suppose that hospitals’ preferences are substitutable. Then there exists a nonempty finite lattice of fixed points

  • X D, X H
  • f Φ which

correspond to stable outcomes A = X D ∩ X H.

Hatfield & Kominers June 4, 2012 13

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Generalized Matching Theory Matching with Contracts

Existence of Stable Allocations

Theorem

Suppose that hospitals’ preferences are substitutable. Then there exists a nonempty finite lattice of fixed points

  • X D, X H
  • f Φ which

correspond to stable outcomes A = X D ∩ X H. The proof follows from the isotonicity of the operator Φ.

Hatfield & Kominers June 4, 2012 13

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Generalized Matching Theory Matching with Contracts

Existence of Stable Allocations

Theorem

Suppose that hospitals’ preferences are substitutable. Then there exists a nonempty finite lattice of fixed points

  • X D, X H
  • f Φ which

correspond to stable outcomes A = X D ∩ X H. The proof follows from the isotonicity of the operator Φ. The lattice result implies opposition of interests.

Hatfield & Kominers June 4, 2012 13

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Generalized Matching Theory Matching with Contracts

The Law of Aggregate Demand

Definition

The preferences of h ∈ H satisfy the Law of Aggregate Demand (LoAD) if for all Y ′ ⊆ Y ⊆ X,

  • C h (Y )
  • ≥
  • C h (Y ′)
  • .

Hatfield & Kominers June 4, 2012 14

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Generalized Matching Theory Matching with Contracts

The Law of Aggregate Demand

Definition

The preferences of h ∈ H satisfy the Law of Aggregate Demand (LoAD) if for all Y ′ ⊆ Y ⊆ X,

  • C h (Y )
  • ≥
  • C h (Y ′)
  • .

Hatfield & Kominers June 4, 2012 14

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Generalized Matching Theory Matching with Contracts

The Law of Aggregate Demand

Definition

The preferences of h ∈ H satisfy the Law of Aggregate Demand (LoAD) if for all Y ′ ⊆ Y ⊆ X,

  • C h (Y )
  • ≥
  • C h (Y ′)
  • .

Intuition: When h receives new offers, he hires at least as many doctors as he did before: no doctor can do the work of two.

Hatfield & Kominers June 4, 2012 14

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Generalized Matching Theory Matching with Contracts

The Rural Hospitals Theorem and Strategy-Proofness

Theorem

If all hospitals’ preferences are substitutable and satisfy the LoAD, then each doctor and hospital signs the same number of contracts at each stable outcome.

Hatfield & Kominers June 4, 2012 15

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Generalized Matching Theory Matching with Contracts

The Rural Hospitals Theorem and Strategy-Proofness

Theorem

If all hospitals’ preferences are substitutable and satisfy the LoAD, then each doctor and hospital signs the same number of contracts at each stable outcome.

Theorem

If all hospitals’ preferences are substitutable and satisfy the LoAD, the doctor-optimal stable many-to-one matching mechanism is (group) strategy-proof.

Hatfield & Kominers June 4, 2012 15

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Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Hatfield & Kominers June 4, 2012 16

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Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Substitutability is sufficient, but is it “necessary”?

Hatfield & Kominers June 4, 2012 16

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Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Substitutability is sufficient, but is it “necessary”? No: Hatfield and Kojima (2010) showed that a weaker condition, bilateral substitutability, is sufficient.

Hatfield & Kominers June 4, 2012 16

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Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Substitutability is sufficient, but is it “necessary”? No: Hatfield and Kojima (2010) showed that a weaker condition, bilateral substitutability, is sufficient.

In simple many-to-one matching, substitutability is necessary.

Hatfield & Kominers June 4, 2012 16

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Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Substitutability is sufficient, but is it “necessary”? No: Hatfield and Kojima (2010) showed that a weaker condition, bilateral substitutability, is sufficient.

In simple many-to-one matching, substitutability is necessary.

This has important applications: S¨

  • nmez and Switzer (2011),

S¨

  • nmez (2011) consider the matching of cadets to U.S. Army

branches, where preferences are not substitutable, but are unilaterally substitutable.

Hatfield & Kominers June 4, 2012 16

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SLIDE 56

Generalized Matching Theory Matching with Contracts

Matching Without Substitutes

Substitutability is sufficient, but is it “necessary”? No: Hatfield and Kojima (2010) showed that a weaker condition, bilateral substitutability, is sufficient.

In simple many-to-one matching, substitutability is necessary.

This has important applications: S¨

  • nmez and Switzer (2011),

S¨

  • nmez (2011) consider the matching of cadets to U.S. Army

branches, where preferences are not substitutable, but are unilaterally substitutable. Open question: What is the necessary and sufficient condition for matching with contracts?

Hatfield & Kominers June 4, 2012 16

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Generalized Matching Theory Supply Chain Matching

Supply Chain Matching (Ostrovsky, 2008) s

  • i
  • b1

b2

Same-side contracts are substitutes. Cross-side contracts are complements. ⇒ Objects are fully substitutable.

Hatfield & Kominers June 4, 2012 17

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Generalized Matching Theory Supply Chain Matching

Supply Chain Matching (Ostrovsky, 2008) s

  • i
  • b1

b2

Same-side contracts are substitutes. Cross-side contracts are complements. ⇒ Objects are fully substitutable.

Theorem

Stable outcomes exist.

Hatfield & Kominers June 4, 2012 17

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Generalized Matching Theory Supply Chain Matching

Full Substitutability is Essential (Hatfield–Kominers, 2012)

Although (full) substitutability is not necessary for many-to-one matching with contracts, it is necessary for

supply chain matching, and many-to-many matching with contracts.

Hatfield & Kominers June 4, 2012 18

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Generalized Matching Theory Supply Chain Matching

Full Substitutability is Essential (Hatfield–Kominers, 2012)

Although (full) substitutability is not necessary for many-to-one matching with contracts, it is necessary for

supply chain matching, and many-to-many matching with contracts.

This poses a problem for couples matching.

Hatfield & Kominers June 4, 2012 18

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Generalized Matching Theory Supply Chain Matching

Full Substitutability is Essential (Hatfield–Kominers, 2012)

Although (full) substitutability is not necessary for many-to-one matching with contracts, it is necessary for

supply chain matching, and many-to-many matching with contracts.

This poses a problem for couples matching. But new large-market results may provide a partial solution:

Kojima–Pathak–Roth (2011); Ashlagi–Braverman–Hassidim (2011); Azevedo–Weyl–White (2012); Azevedo–Hatfield (in preparation).

Hatfield & Kominers June 4, 2012 18

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Generalized Matching Theory Supply Chain Matching

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Hatfield & Kominers June 4, 2012 19

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Generalized Matching Theory Supply Chain Matching

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Theorem

Acyclicity is necessary for stability.

Hatfield & Kominers June 4, 2012 19

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Generalized Matching Theory Supply Chain Matching

The Rural Hospitals Theorem

Theorem (two-sided)

In many-to-one (or -many) matching with contracts, if all preferences are substitutable and satisfy the LoAD, then each doctor and hospital signs the same number of contracts at each stable outcome.

Hatfield & Kominers June 4, 2012 20

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SLIDE 65

Generalized Matching Theory Supply Chain Matching

The Rural Hospitals Theorem

Theorem (two-sided)

In many-to-one (or -many) matching with contracts, if all preferences are substitutable and satisfy the LoAD, then each doctor and hospital signs the same number of contracts at each stable outcome. What happens in supply chains?

s

x

  • z
  • i

y

  • b

Ps : {x} ≻ {z} ≻ ∅ Pi : {x, y} ≻ ∅ Pb : {z} ≻ {x} ≻ ∅

Hatfield & Kominers June 4, 2012 20

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Generalized Matching Theory Supply Chain Matching

The Rural Hospitals Theorem

Theorem (two-sided)

In many-to-one (or -many) matching with contracts, if all preferences are substitutable and satisfy the LoAD, then each doctor and hospital signs the same number of contracts at each stable outcome.

Theorem (supply chain)

Suppose that X is acyclic and that all preferences are fully substitutable and satisfy LoAD (and LoAS). Then, for each agent f ∈ F, the difference between the number of contracts the f buys and the number of contracts f sells is invariant across stable outcomes.

Hatfield & Kominers June 4, 2012 20

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SLIDE 67

Generalized Matching Theory The Assignment Problem

The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972)

ζm,w ∼ total surplus of marriage of man m and woman w

Hatfield & Kominers June 4, 2012 21

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SLIDE 68

Generalized Matching Theory The Assignment Problem

The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972)

ζm,w ∼ total surplus of marriage of man m and woman w assignment indicators: am,w ≡

  • 1

m, w married

  • therwise

Hatfield & Kominers June 4, 2012 21

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Generalized Matching Theory The Assignment Problem

The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972)

ζm,w ∼ total surplus of marriage of man m and woman w assignment indicators: am,w ≡

  • 1

m, w married

  • therwise

Stable assignment (˜ am,w) solves the integer program max

  • m
  • w

am,wζm,w

  • 0 ≤

w am,w ≤ 1

∀m 0 ≤

m am,w ≤ 1

∀w

Hatfield & Kominers June 4, 2012 21

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Generalized Matching Theory The Assignment Problem

“Efficient Mating”

zm,w ≡ ζm,w − ζm,∅ − ζ∅,w ∼ marital surplus max

  • m
  • w

am,wζm,w = max

  • m
  • w

am,wzm,w +

  • m

ζm,∅ +

  • w

ζ∅,w

  • Theorem

Stable assignment maximizes aggregate marriage output.

Note

Even with am,w ∈ [0, 1], the optimum is always an integer solution.

Hatfield & Kominers June 4, 2012 22

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SLIDE 71

Generalized Matching Theory The Assignment Problem

Other Notes

Dual problem shows us “shadow prices” which describe the social cost of removing an agent from the pool of singles. If ζm,w = h(xm, yw), then complementarity (substitution) in traits leads to positive (negative) assortative mating. (Becker, 1973) Matches stable in the presence of transfers need not be stable if transfers are not allowed, and vice versa. (Jaffe–Kominers, tomorrow)

Hatfield & Kominers June 4, 2012 23

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Generalization to Networks

Main Results

In arbitrary trading networks with

1 bilateral contracts, 2 transferable utility, and 3 fully substitutable preferences,

competitive equilibria exist and coincide with stable outcomes.

Hatfield & Kominers June 4, 2012 24

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SLIDE 73

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Generalization to Networks

Main Results

In arbitrary trading networks with

1 bilateral contracts, 2 transferable utility, and 3 fully substitutable preferences,

competitive equilibria exist and coincide with stable outcomes. Full substitutability is necessary for these results. Correspondence results extend to other solutions concepts.

Hatfield & Kominers June 4, 2012 24

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SLIDE 74

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Theorem

Acyclicity is necessary for stability!

Hatfield & Kominers June 4, 2012 25

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SLIDE 75

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Related Literature

Matching: Kelso–Crawford (1982): Many-to-one (with transfers); (GS) Ostrovsky (2008): Supply chain networks; (SSS) and (CSC) Hatfield–Kominers (2012): Trading networks (sans transfers) Exchange economies with indivisibilities: Koopmans–Beckmann (1957); Shapley–Shubik (1972) Gul–Stachetti (1999): (GS) Sun–Yang (2006, 2009): (GSC)

Hatfield & Kominers June 4, 2012 26

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SLIDE 76

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Setting: Trades and Contracts

Finite set of agents I

Hatfield & Kominers June 4, 2012 27

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SLIDE 77

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Setting: Trades and Contracts

Finite set of agents I Finite set of bilateral trades Ω

each trade ω ∈ Ω has a seller s(ω) ∈ I and a buyer b(ω) ∈ I

An arrangement is a pair [Ψ; p], where Ψ ⊆ Ω and p ∈ R|Ω|.

Hatfield & Kominers June 4, 2012 27

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SLIDE 78

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Setting: Trades and Contracts

Finite set of agents I Finite set of bilateral trades Ω

each trade ω ∈ Ω has a seller s(ω) ∈ I and a buyer b(ω) ∈ I

An arrangement is a pair [Ψ; p], where Ψ ⊆ Ω and p ∈ R|Ω|. Set of contracts X := Ω × R

each contract x ∈ X is a pair (ω, pω) τ(Y ) ⊆ Ω ∼ set of trades in contract set Y ⊆ X

A (feasible) outcome is a set of contracts A ⊆ X which uniquely prices each trade in A.

Hatfield & Kominers June 4, 2012 27

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Setting: Demand

Each agent i has quasilinear utility over arrangements: Ui ([Ψ; p]) = ui(Ψi) +

  • ψ∈Ψi→

pψ −

  • ψ∈Ψ→i

pψ. Ui extends naturally to (feasible) outcomes.

Hatfield & Kominers June 4, 2012 28

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Setting: Demand

Each agent i has quasilinear utility over arrangements: Ui ([Ψ; p]) = ui(Ψi) +

  • ψ∈Ψi→

pψ −

  • ψ∈Ψ→i

pψ. Ui extends naturally to (feasible) outcomes. For any price vector p ∈ R|Ω|, the demand of i is Di(p) = argmaxΨ⊆Ωi Ui([Ψ; p]). For any set of contracts Y ⊆ X, the choice of i is Ci(Y ) = argmaxZ⊆Yi Ui(Z).

Hatfield & Kominers June 4, 2012 28

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Assumptions on Preferences

1 ui(Ψ) ∈ R ∪ {−∞}. Hatfield & Kominers June 4, 2012 29

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SLIDE 82

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Assumptions on Preferences

1 ui(Ψ) ∈ R ∪ {−∞}. 2 ui(∅) ∈ R. Hatfield & Kominers June 4, 2012 29

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SLIDE 83

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Assumptions on Preferences

1 ui(Ψ) ∈ R ∪ {−∞}. 2 ui(∅) ∈ R. 3 Full substitutability... Hatfield & Kominers June 4, 2012 29

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Full Substitutability (I)

Definition

The preferences of agent i are fully substitutable (in choice language) if

1 same-side contracts are substitutes for i, and 2 cross-side contracts are complements for i. Hatfield & Kominers June 4, 2012 30

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Full Substitutability (I)

Definition

The preferences of agent i are fully substitutable (in choice language) if for all sets of contracts Y , Z ⊆ Xi such that |Ci(Z)| = |Ci(Y )| = 1,

1 if Yi→ = Zi→, and Y→i ⊆ Z→i, then for Y ∗ ∈ Ci(Y ) and

Z ∗ ∈ Ci(Z), we have (Y→i − Y ∗

→i) ⊆ (Z→i − Z ∗ →i) and

Y ∗

i→ ⊆ Z ∗ i→;

2 if Y→i = Z→i, and Yi→ ⊆ Zi→, then for Y ∗ ∈ Ci(Y ) and

Z ∗ ∈ Ci(Z), we have (Yi→ − Y ∗

i→) ⊆ (Zi→ − Z ∗ i→) and

Y ∗

→i ⊆ Z ∗ →i.

Hatfield & Kominers June 4, 2012 30

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Full Substitutability (II)

Theorem

Choice-language full substitutability

1 has equivalents in demand and “indicator” languages; 2 holds if and only if the indirect utility function

Vi(p) := max

Ψ⊆Ωi Ui([Ψ; p])

is submodular (Vi(p ∨ q) + Vi(p ∧ q) ≤ Vi(p) + Vi(q)).

Hatfield & Kominers June 4, 2012 31

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Solution Concepts

Definition

An outcome A is stable if it is

1 Individually rational: for each i ∈ I, Ai ∈ Ci(A); 2 Unblocked: There is no nonempty, feasible Z ⊆ X such that

Z ∩ A = ∅ and for each i, and for each Yi ∈ Ci(Z ∪ A), we have Zi ⊆ Yi.

Definition

Arrangement [Ψ; p] is a competitive equilibrium (CE) if for each i, Ψi ∈ Di(p).

Hatfield & Kominers June 4, 2012 32

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Existence of Competitive Equilibria

Theorem

If preferences are fully substitutable, then a CE exists.

Proof

1 Modify: Transform potentially unbounded ui to ˆ

ui.

2 Associate: Construct a two-sided one-to-many matching market:

     i → “firm”: valuation ˜ ui(Ψ) := ˆ ui(Ψ→i ∪ (Ω − Ψ)i→); ω → “worker”: wants high wages; p → “wage”.

3 A CE exists in the associated market (Kelso–Crawford, 1982). 4 CE associated → CE modified = CE original. Hatfield & Kominers June 4, 2012 33

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Structure of Competitive Equilibria

Theorem (First Welfare Theorem)

Let [Ψ; p] be a CE. Then Ψ is efficient.

Hatfield & Kominers June 4, 2012 34

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Structure of Competitive Equilibria

Theorem (First Welfare Theorem)

Let [Ψ; p] be a CE. Then Ψ is efficient.

Theorem (Second Welfare Theorem)

Suppose agents’ preferences are fully substitutable. Then, for any CE [Ξ; p] and efficient set of trades Ψ, [Ψ; p] is a CE.

Hatfield & Kominers June 4, 2012 34

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Structure of Competitive Equilibria

Theorem (First Welfare Theorem)

Let [Ψ; p] be a CE. Then Ψ is efficient.

Theorem (Second Welfare Theorem)

Suppose agents’ preferences are fully substitutable. Then, for any CE [Ξ; p] and efficient set of trades Ψ, [Ψ; p] is a CE.

Theorem (Lattice Structure)

The set of CE price vectors is a lattice.

Hatfield & Kominers June 4, 2012 34

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Relationship Between Stability and CE

Theorem

If [Ψ; p] is a CE, then A ≡ ∪ψ∈Ψ{(ψ, pψ)} is stable. The reverse implication is not true in general.

Hatfield & Kominers June 4, 2012 35

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

The Relationship Between Stability and CE

Theorem

If [Ψ; p] is a CE, then A ≡ ∪ψ∈Ψ{(ψ, pψ)} is stable. The reverse implication is not true in general.

Theorem

Suppose that agents’ preferences are fully substitutable and A is

  • stable. Then, there exists a price vector p ∈ R|Ω| such that

1 [τ(A); p] is a CE, and 2 if (ω, ¯

pω) ∈ A, then pω = ¯ pω.

Hatfield & Kominers June 4, 2012 35

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Full Substitutability is Necessary

Theorem

Suppose that there exist at least four agents and that the set of trades is exhaustive. Then, if the preferences of some agent i are not fully substitutable, there exist “simple” preferences for all agents j = i such that no stable outcome exists.

Hatfield & Kominers June 4, 2012 36

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Full Substitutability is Necessary

Theorem

Suppose that there exist at least four agents and that the set of trades is exhaustive. Then, if the preferences of some agent i are not fully substitutable, there exist “simple” preferences for all agents j = i such that no stable outcome exists.

Corollary

Under the conditions of the above theorem, there exist “simple” preferences for all agents j = i such that no CE exists.

Hatfield & Kominers June 4, 2012 36

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Alternative Solution Concepts

Definition

An outcome A is in the core if there is no group deviation Z such that Ui(Z) > Ui(A) for all i associated with Z.

Hatfield & Kominers June 4, 2012 37

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Alternative Solution Concepts

Definition

An outcome A is in the core if there is no group deviation Z such that Ui(Z) > Ui(A) for all i associated with Z.

Definition

A set of contracts Z is a chain if its elements can be arranged in some

  • rder y1, . . . , y|Z| such that s(yℓ+1) = b(yℓ) for all ℓ < |Z|.

Hatfield & Kominers June 4, 2012 37

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SLIDE 98

Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Alternative Solution Concepts

Definition

An outcome A is in the core if there is no group deviation Z such that Ui(Z) > Ui(A) for all i associated with Z.

Definition

A set of contracts Z is a chain if its elements can be arranged in some

  • rder y1, . . . , y|Z| such that s(yℓ+1) = b(yℓ) for all ℓ < |Z|.

Definition

Outcome A is stable if it is individually rational and Unblocked: There is no nonempty, feasible Z ⊆ X such that

Z ∩ A = ∅ and for each i, and for each Yi ∈ Ci(Z ∪ A), we have Zi ⊆ Yi.

Hatfield & Kominers June 4, 2012 37

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Alternative Solution Concepts

Definition

An outcome A is in the core if there is no group deviation Z such that Ui(Z) > Ui(A) for all i associated with Z.

Definition

A set of contracts Z is a chain if its elements can be arranged in some

  • rder y1, . . . , y|Z| such that s(yℓ+1) = b(yℓ) for all ℓ < |Z|.

Definition

Outcome A is chain stable if it is individually rational and Unblocked: There is no nonempty, feasible chain Z ⊆ X s.t.

Z ∩ A = ∅ and for each i, and for each Yi ∈ Ci(Z ∪ A), we have Zi ⊆ Yi.

Hatfield & Kominers June 4, 2012 37

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Alternative Solution Concepts

Definition

An outcome A is in the core if there is no group deviation Z such that Ui(Z) > Ui(A) for all i associated with Z.

Definition

A set of contracts Z is a chain if its elements can be arranged in some

  • rder y1, . . . , y|Z| such that s(yℓ+1) = b(yℓ) for all ℓ < |Z|.

Definition

Outcome A is strongly group stable if it is individually rational and Unblocked: There is no nonempty, feasible Z ⊆ X such that

Z ∩ A = ∅ and for each i associated with Z, there exists a Y i ⊆ Z ∪ A such that Zi ⊆ Y i and Ui(Y i) > Ui(A).

Hatfield & Kominers June 4, 2012 37

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Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks

Relationship Between the Concepts

CE

  • Strongly Group Stable
  • Stable
  • Chain Stable
  • Core
  • Efficient

Hatfield & Kominers June 4, 2012 38

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SLIDE 102

Generalized Matching Theory Multilateral Matching

Multilateral Contracts

Cu

  • Sn
  • Bronzemaker

Toymaker Full substitutability is “necessary” in (Discrete, Bilateral) Contract Matching with Transfers.

Hatfield & Kominers June 4, 2012 39

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SLIDE 103

Generalized Matching Theory Multilateral Matching

Multilateral Contracts

Cu

  • (ψ,rψ,sψ)
  • Sn
  • (ϕ,rϕ,sϕ)
  • (ω, rω, sω)
  • Bronzemaker

Toymaker Full substitutability is “necessary” in (Discrete, Bilateral) Contract Matching with Transfers.

Hatfield & Kominers June 4, 2012 39

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SLIDE 104

Generalized Matching Theory Multilateral Matching

Multilateral Contracts

Publisher 1

  • (ψ,rψ,sψ)
  • Publisher 2
  • (ϕ,rϕ,sϕ)
  • (ω, rω, sω)
  • Ad Exchange

Residual Networks Full substitutability is “necessary” in (Discrete, Bilateral) Contract Matching with Transfers.

Hatfield & Kominers June 4, 2012 39

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SLIDE 105

Generalized Matching Theory Multilateral Matching

Multilateral Contracts

Main Results

In arbitrary trading networks with

1 multilateral contracts, 2 transferable utility, 3 concave preferences, and 4 continuously divisible contracts,

competitive equilibria exist and coincide with stable outcomes. = ⇒ Some production complementarities “work” in matching!

Hatfield & Kominers June 4, 2012 39

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SLIDE 106

Generalized Matching Theory Conclusion

Discussion

Applications of stability in absence of CE? Linear programming approach? Empirical applications? Substitutability vs. concavity?

Hatfield & Kominers June 4, 2012 40

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SLIDE 107

Generalized Matching Theory Conclusion

Discussion

Applications of stability in absence of CE? Linear programming approach? Empirical applications? Substitutability vs. concavity? \end{Talk}

Hatfield & Kominers June 4, 2012 40

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SLIDE 108

Extra Slides

Demand-Language Full Substitutability

Definition

The preferences of agent i are fully substitutable in demand language if for all p, p′ ∈ R|Ω| such that |Di(p)| = |Di(p′)| = 1,

1 if pω = p′

ω for all ω ∈ Ωi→, and pω ≥ p′ ω for all ω ∈ Ω→i, then

for the unique Ψ ∈ Di(p) and Ψ′ ∈ Di(p′), we have Ψi→ ⊆ Ψ′

i→,

{ω ∈ Ψ′

→i : pω = p′ ω} ⊆ Ψ→i;

2 if pω = p′

ω for all ω ∈ Ω→i, and pω ≤ p′ ω for all ω ∈ Ωi→, then

for the unique Ψ ∈ Di(p) and Ψ′ ∈ Di(p′), we have Ψ→i ⊆ Ψ′

→i,

{ω ∈ Ψ′

i→ : pω = p′ ω} ⊆ Ψi→.

Hatfield & Kominers June 4, 2012 41

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SLIDE 109

Extra Slides

Indicator-Language Full Substitutability

ei

ω(Ψ) =

     1 ω ∈ Ψ→i −1 ω ∈ Ψi→

  • therwise

Definition

The preferences of agent i are fully substitutable in indicator language if for all price vectors p, p′ ∈ R|Ω| such that |Di(p)| = |Di(p′)| = 1 and p ≤ p′, for Ψ ∈ Di(p) and Ψ′ ∈ Di(p′), we have ei

ω(Ψ) ≤ ei ω(Ψ′)

for each ω ∈ Ωi such that pω = p′

ω.

Hatfield & Kominers June 4, 2012 42