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Frontiers of Matching Theory Scott Duke Kominers Department of Economics, Harvard University, and Harvard Business School Colloquium Department of Mathematics, Vassar College October 12, 2010 Scott Duke Kominers (Harvard) October 12, 2010 1

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Frontiers of Matching Theory

Scott Duke Kominers

Department of Economics, Harvard University, and Harvard Business School

Colloquium

Department of Mathematics, Vassar College

October 12, 2010

Scott Duke Kominers (Harvard) October 12, 2010 1

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with 1 man and 0 women, how can we arrange marriages so that there are no divorces? m1 w1

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with 1 man and 1 woman, how can we arrange marriages so that there are no divorces? m1 w1

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with 1 man and 1 woman, how can we arrange marriages so that there are no divorces? m1 w1

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with 1 man and 1 woman, how can we arrange marriages so that there are no divorces? m1 w1

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with 3 men and 1 woman, how can we arrange marriages so that there are no divorces? m1 w1 m2 m3

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with M men and 1 woman, how can we arrange marriages so that there are no divorces? m1 w . . . mM

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

The Marriage Problem (Gale–Shapley, 1962)

Question

In a society with M men and W women, how can we arrange marriages so that there are no divorces? m1 w1 . . . . . . mM wW

Scott Duke Kominers (Harvard) October 12, 2010 2

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Matching Theory Introduction

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.

Scott Duke Kominers (Harvard) October 12, 2010 3

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Matching Theory Introduction

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 his next-highest choice woman. 2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

Scott Duke Kominers (Harvard) October 12, 2010 3

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Matching Theory Introduction

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 his next-highest choice woman. 2 Each woman holds onto her most-preferred acceptable proposal

(if any) and rejects all others.

At termination, no agent wants a divorce!

Scott Duke Kominers (Harvard) October 12, 2010 3

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Matching Theory Stable Marriage

Stability

Definition

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

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Stability

Definition

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

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Stability

Definition

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

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Stability

Definition

A marriage matching µ is stable if no agent wants a divorce: Rational: All agents i find their matches µ(i) acceptable.

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Stability

Definition

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

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Stability

Definition

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

Theorem (Gale–Shapley, 1962)

A stable marriage matching exists.

Scott Duke Kominers (Harvard) October 12, 2010 4

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Matching Theory Stable Marriage

Lattice Structure: Opposition of Interests

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.

Scott Duke Kominers (Harvard) October 12, 2010 5

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Matching Theory Stable Marriage

Lattice Structure: Opposition of Interests

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.

Scott Duke Kominers (Harvard) October 12, 2010 5

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Matching Theory Stable Marriage

Lattice Structure: Opposition of Interests

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.

Scott Duke Kominers (Harvard) October 12, 2010 5

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Matching Theory Stable Marriage

Opposition of Interests: A Simple Example

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

Scott Duke Kominers (Harvard) October 12, 2010 6

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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

Scott Duke Kominers (Harvard) October 12, 2010 6

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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

Scott Duke Kominers (Harvard) October 12, 2010 6

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Proof

¯ µ = man-optimal stable match; µ = any stable match

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Proof

¯ µ = man-optimal stable match; µ = any stable match ¯ µM ¯ µW µM µW

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Proof

¯ µ = man-optimal stable match; µ = any stable match ¯ µM ¯ µW ⊆ µM µW

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Proof

¯ µ = man-optimal stable match; µ = any stable match ¯ µM ¯ µW ⊆ ⊇ µM µW

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory Stable Marriage

The “Lone Wolf” Theorem (Roth, 1984)

Theorem

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

Proof

¯ µ = man-optimal stable match; µ = any stable match ¯ µM

card

= ¯ µW ⊆ ⊇ µM

card

= µW

Scott Duke Kominers (Harvard) October 12, 2010 7

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Matching Theory

Generalizations

1962: Many-to-one Matching (“College Admissions”)

Substitutable preferences sufficient for stability “Rural Hospitals” Theorem

Scott Duke Kominers (Harvard) October 12, 2010 8

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Matching Theory

Generalizations

1962: Many-to-one Matching (“College Admissions”)

Substitutable preferences sufficient for stability “Rural Hospitals” Theorem

1985±ε: Many-to-many Matching (“Consultants and Firms”)

Multiple notions of stability

Scott Duke Kominers (Harvard) October 12, 2010 8

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Matching Theory

Generalizations

1962: Many-to-one Matching (“College Admissions”)

Substitutable preferences sufficient for stability “Rural Hospitals” Theorem

1985±ε: Many-to-many Matching (“Consultants and Firms”)

Multiple notions of stability

2005: Matching with Contracts (“Doctors and Hospitals”)

{Wage, schedule, . . .} negotiations embed into matching

Scott Duke Kominers (Harvard) October 12, 2010 8

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Generalized Matching Theory Matching with Contracts

Matching with Contracts (Hatfield–Milgrom, 2005)

x = (doctor, hospital, terms)

Scott Duke Kominers (Harvard) October 12, 2010 9

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Generalized Matching Theory Matching with Contracts

Matching with Contracts (Hatfield–Milgrom, 2005)

X ⊆ D × H × T x = (doctor, hospital, terms)

Scott Duke Kominers (Harvard) October 12, 2010 9

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Generalized Matching Theory Matching with Contracts

(Many-to-one) Matching with Contracts (Hatfield–Milgrom)

X ⊆ D × H × T x = (doctor, hospital, terms)

Scott Duke Kominers (Harvard) October 12, 2010 9

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Generalized Matching Theory Matching with Contracts

(Many-to-one) Matching with Contracts (Hatfield–Milgrom)

X ⊆ D × H × T x = (doctor, hospital, terms) Assumptions

Hospitals have strict preferences over sets of contracts. Doctors have strict preferences and “unit demand.”

Scott Duke Kominers (Harvard) October 12, 2010 9

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Generalized Matching Theory Matching with Contracts

(Many-to-one) Matching with Contracts (Hatfield–Milgrom)

X ⊆ D × H × T x = (doctor, hospital, terms) Special Cases

Men–Women (X = M × W × {1}; all have unit demand) Colleges–Students (X = S × C × {1})

Scott Duke Kominers (Harvard) October 12, 2010 9

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Generalized Matching Theory Matching with Contracts

Substitutability

Definition

The preferences of an agent f ∈ D ∪ H are substitutable if there do not exist x, z ∈ X and Y ⊆ X such that z / ∈ C f (Y ∪ {z}) but z ∈ C f (Y ∪ {x, z}) .

Scott Duke Kominers (Harvard) October 12, 2010 10

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Generalized Matching Theory Matching with Contracts

Substitutability

Definition

The preferences of an agent f ∈ D ∪ H are substitutable if there do not exist x, z ∈ X and Y ⊆ X such that z / ∈ C f (Y ∪ {z}) but z ∈ C f (Y ∪ {x, z}) .

Intuition

Receiving new offers makes f (weakly) less interested in old offers.

Scott Duke Kominers (Harvard) October 12, 2010 10

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Generalized Matching Theory Matching with Contracts

Substitutability

Definition

The preferences of an agent f ∈ D ∪ H are substitutable if there do not exist x, z ∈ X and Y ⊆ X such that z / ∈ C f (Y ∪ {z}) but z ∈ C f (Y ∪ {x, z}) .

Intuition

Receiving new offers makes f (weakly) less interested in old offers.

Equivalent Definition

The rejection function Rf (X ′) = X ′ − C f (X ′) is monotone.

Scott Duke Kominers (Harvard) October 12, 2010 10

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Generalized Matching Theory Matching with Contracts

Substitutability ⇒ Stability

Theorem

Suppose that all preferences are substitutable. Then, the set of stable allocations is a nonempty lattice.

Scott Duke Kominers (Harvard) October 12, 2010 11

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Generalized Matching Theory Matching with Contracts

Substitutability ⇒ Stability

Theorem

Suppose that all preferences are substitutable. Then, the set of stable allocations is a nonempty lattice.

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y ))

Scott Duke Kominers (Harvard) October 12, 2010 11

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Generalized Matching Theory Matching with Contracts

Substitutability ⇒ Stability

Theorem

Suppose that all preferences are substitutable. Then, the set of stable allocations is a nonempty lattice.

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D(Y ).

Scott Duke Kominers (Harvard) October 12, 2010 11

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Generalized Matching Theory Matching with Contracts

Substitutability ⇒ Stability

Theorem

Suppose that all preferences are substitutable. Then, the set of stable allocations is a nonempty lattice.

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D(Y ). If RH and RD are monotone, then Φ is monotone.

Scott Duke Kominers (Harvard) October 12, 2010 11

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Generalized Matching Theory Matching with Contracts

Substitutability ⇒ Stability

Theorem

Suppose that all preferences are substitutable. Then, the set of stable allocations is a nonempty lattice.

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D(Y ). If RH and RD are monotone, then Φ is monotone. Tarski’s Fixed Point Theorem = ⇒ a lattice of fixed points of Φ.

Scott Duke Kominers (Harvard) October 12, 2010 11

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Frontiers of Matching Theory

How deep is the rabbit hole?

Question What is “needed” in order for matching theory to work?

Scott Duke Kominers (Harvard) October 12, 2010 12

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Frontiers of Matching Theory Matching in Networks

Matching in Networks (Hatfield–K., 2010) x = (buyer, seller, terms)

Scott Duke Kominers (Harvard) October 12, 2010 13

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Frontiers of Matching Theory Matching in Networks

Matching in Networks (Hatfield–K., 2010) X ⊆ F × F × T x = (buyer, seller, terms)

Scott Duke Kominers (Harvard) October 12, 2010 13

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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SLIDE 52

Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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SLIDE 57

Frontiers of Matching Theory Matching in Networks

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in 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!

Scott Duke Kominers (Harvard) October 12, 2010 14

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Frontiers of Matching Theory Matching in Networks

Matching in Networks (Hatfield–K., 2010) X ⊆ F × F × T x = (buyer, seller, terms)

Scott Duke Kominers (Harvard) October 12, 2010 15

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Frontiers of Matching Theory Matching in Networks

Matching in Networks (Hatfield–K., 2010) X ⊆ F × F × T x = (buyer, seller, terms)

Assumptions

Agents have strict preferences over sets of contracts. The contract graph is acyclic ( ⇐ ⇒ supply chain structure).

Scott Duke Kominers (Harvard) October 12, 2010 15

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Frontiers of Matching Theory Matching in Networks

Matching in Networks (Hatfield–K., 2010) X ⊆ F × F × T x = (buyer, seller, terms)

Special Cases

Doctors–Hospitals (X ⊆ D × H × T) Supply chain Matching

Scott Duke Kominers (Harvard) October 12, 2010 15

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Frontiers of Matching Theory Matching in Networks

Stability

Definition

An allocation of contracts A is stable if no set of agents (strictly) prefers to match among themselves than to accept the terms of A. That is, A is stable if it is

1 Rational 2 Unblocked Scott Duke Kominers (Harvard) October 12, 2010 16

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Frontiers of Matching Theory Matching in Networks

Stability

Definition

An allocation of contracts A is stable if no set of agents (strictly) prefers to match among themselves than to accept the terms of A. Formally: A is stable if it is

1 Rational: For all f ∈ F, C f (A) = A|f . 2 Unblocked: There does not exist a nonempty blocking set

Z ⊆ X such that Z ∩ A = ∅ and Z|f ⊆ C f (A ∪ Z) (for all f ).

Scott Duke Kominers (Harvard) October 12, 2010 16

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Frontiers of Matching Theory Matching in Networks

Substitutability

Definition

The preferences of an agent f are fully substitutable if receiving more buyer (seller) contracts makes f weakly less interested in his available buyer (seller) contracts and weakly more interested in his available seller (buyer) contracts.

Intuition

same-side contracts are substitutes cross-side contracts are complements

Scott Duke Kominers (Harvard) October 12, 2010 17

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Frontiers of Matching Theory Matching in Networks

Full Substitutability ⇐ ⇒ Guaranteed Stability

Theorem (Sufficiency)

If X is acyclic and all preferences are fully substitutable, then there exists a lattice of stable allocations.

Scott Duke Kominers (Harvard) October 12, 2010 18

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Frontiers of Matching Theory Matching in Networks

Full Substitutability ⇐ ⇒ Guaranteed Stability

Theorem (Sufficiency)

If X is acyclic and all preferences are fully substitutable, then there exists a lattice of stable allocations.

Theorem (Necessity)

Both conditions in the above theorem are necessary for the result.

Scott Duke Kominers (Harvard) October 12, 2010 18

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Frontiers of Matching Theory

Extensions

Scott Duke Kominers (Harvard) October 12, 2010 19

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Frontiers of Matching Theory

Extensions

Surprising generalization of “Lone Wolf” Theorem

Agents’ excess stocks are invariant

Scott Duke Kominers (Harvard) October 12, 2010 19

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Frontiers of Matching Theory

Extensions

Surprising generalization of “Lone Wolf” Theorem

Agents’ excess stocks are invariant

Design of contract language

Available contract set affects outcomes

Scott Duke Kominers (Harvard) October 12, 2010 19

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Frontiers of Matching Theory

Extensions

Surprising generalization of “Lone Wolf” Theorem

Agents’ excess stocks are invariant

Design of contract language

Available contract set affects outcomes

Completion of many-to-one preferences

New conditions sufficient for many-to-one stability

Scott Duke Kominers (Harvard) October 12, 2010 19

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Frontiers of Matching Theory

Extensions

Surprising generalization of “Lone Wolf” Theorem

Agents’ excess stocks are invariant

Design of contract language

Available contract set affects outcomes

Completion of many-to-one preferences

New conditions sufficient for many-to-one stability

Matching with money

Pigouvian taxes restore stability for cyclic X

Scott Duke Kominers (Harvard) October 12, 2010 19

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Frontiers of Matching Theory

The Law of Aggregate Demand

Definition

Preferences of f satisfy the Law of Aggregate Demand (LoAD) if, whenever f receives new offers as a buyer, he takes on at least as many new buyer contracts he does seller contracts.

Intuition

When f buys a new good, he will sell at most one more good than he was previously selling. Law of Aggregate Supply (LoAS) is analogous.

Scott Duke Kominers (Harvard) October 12, 2010 20

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Frontiers of Matching Theory

The Law of Aggregate Demand

Definition

Preferences of f satisfy the Law of Aggregate Demand (LoAD) if, whenever f receives new offers as a buyer, he takes on at least as many new buyer contracts he does seller contracts. Formally: for all Y , Y ′, Z ⊆ X such that Y ′ ⊆ Y ,

  • C f

B (Y |Z)

  • −
  • C f

B (Y ′|Z)

  • ≥
  • C f

S (Z|Y )

  • −
  • C f

S (Z|Y ′)

  • .

Intuition

When f buys a new good, he will sell at most one more good than he was previously selling. Law of Aggregate Supply (LoAS) is analogous.

Scott Duke Kominers (Harvard) October 12, 2010 20

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Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Scott Duke Kominers (Harvard) October 12, 2010 21

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SLIDE 75

Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Scott Duke Kominers (Harvard) October 12, 2010 21

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SLIDE 76

Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Scott Duke Kominers (Harvard) October 12, 2010 21

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SLIDE 77

Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Theorem (Hatfield–Milgrom, 2005)

In many-to-one matching with contracts: substitutability + LoAD = ⇒ the number of contracts signed by each agent is invariant across stable allocations.

Scott Duke Kominers (Harvard) October 12, 2010 21

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SLIDE 78

Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Theorem (Hatfield–Milgrom, 2005)

In many-to-one matching with contracts: substitutability + LoAD = ⇒ the number of contracts signed by each agent is invariant across stable allocations.

Scott Duke Kominers (Harvard) October 12, 2010 21

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SLIDE 79

Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Theorem (Hatfield–Milgrom, 2005)

In many-to-one matching with contracts: substitutability + LoAD = ⇒ the number of contracts signed by each agent is invariant across stable allocations.

Theorem

Acyclicity + Full Substitutability + LoAD + LoAS = ⇒ each agent holds the same excess stock at every stable allocation.

Scott Duke Kominers (Harvard) October 12, 2010 21

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Frontiers of Matching Theory

The (Generalized) “Lone Wolf” Theorem

Theorem (Roth, 1984)

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

Theorem (Hatfield–Milgrom, 2005)

In many-to-one matching with contracts: substitutability + LoAD = ⇒ the number of contracts signed by each agent is invariant across stable allocations.

Theorem

Acyclicity + Full Substitutability + LoAD + LoAS = ⇒ each agent holds the same excess stock at every stable allocation.

“Matching in Networks with Bilateral Contracts” (Hatfield–K.)

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Frontiers of Matching Theory

Bundling of Contract Terms

1 Work and wages contracted simultaneously:

Employee Preferences: {xw,$} ≻ ∅ Employer Preferences: {xw,$} ≻ ∅

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Frontiers of Matching Theory

Bundling of Contract Terms

1 Work and wages contracted simultaneously:

Employee Preferences: {xw,$} ≻ ∅ Employer Preferences: {xw,$} ≻ ∅

2 Work and wages contracted separately:

Employee Preferences: {x$} ≻ {xw, x$} ≻ ∅

Employer Preferences: {xw} ≻ {xw, x$} ≻ ∅

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SLIDE 83

Frontiers of Matching Theory

Bundling of Contract Terms

1 Work and wages contracted simultaneously:

Employee Preferences: {xw,$} ≻ ∅ Employer Preferences: {xw,$} ≻ ∅

2 Work and wages contracted separately:

Employee Preferences: {x$} ≻ {xw, x$} ≻ ∅

Employer Preferences: {xw} ≻ {xw, x$} ≻ ∅

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SLIDE 84

Frontiers of Matching Theory

Bundling of Contract Terms

1 Work and wages contracted simultaneously:

Employee Preferences: {xw,$} ≻ ∅ Employer Preferences: {xw,$} ≻ ∅

2 Work and wages contracted separately:

Employee Preferences: {x$} ≻ {xw, x$} ≻ ∅

Employer Preferences: {xw} ≻ {xw, x$} ≻ ∅

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SLIDE 85

Frontiers of Matching Theory

Bundling of Contract Terms

1 Work and wages contracted simultaneously:

Employee Preferences: {xw,$} ≻ ∅ Employer Preferences: {xw,$} ≻ ∅

2 Work and wages contracted separately:

Employee Preferences: {x$} ≻ {xw, x$} ≻ ∅

Employer Preferences: {xw} ≻ {xw, x$} ≻ ∅

“Contract Design and Stability in Matching Markets” (Hatfield–K.)

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Frontiers of Matching Theory

Completion of Preferences

Consider the case of one hospital h with preferences ≻h:

  • xα, zβ

≻

  • xβ

≻ {xα} ≻

  • zβ

, which are not substitutable.

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Frontiers of Matching Theory

Completion of Preferences

Consider the case of one hospital h with preferences ≻h:

  • xα, zβ

≻

  • xβ

≻ {xα} ≻

  • zβ

, which are not substitutable. This hospital h actually has preferences ≻h:

  • xα, xβ

≻

  • xα, zβ

≻

  • xβ

≻ {xα} ≻

  • zβ

, which ARE substitutable.

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Frontiers of Matching Theory

Completion of Preferences

Consider the case of one hospital h with preferences ≻h:

  • xα, zβ

≻

  • xβ

≻ {xα} ≻

  • zβ

, which are not substitutable. This hospital h actually has preferences ≻h:

  • xα, xβ

≻

  • xα, zβ

≻

  • xβ

≻ {xα} ≻

  • zβ

, which ARE substitutable.

“Contract Design and Stability in Matching Markets” (Hatfield–K.)

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Frontiers of Matching Theory

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!

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Frontiers of Matching Theory

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Theorem

Acyclicity or transferable utility is necessary for stability!

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Frontiers of Matching Theory

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Theorem

Acyclicity or transferable utility is necessary for stability!

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Frontiers of Matching Theory

Cyclic Contract Sets g f1

y

  • x1
  • f2

x2

  • Pf1 : {y, x2} ≻ {x1, x2} ≻ ∅

Pf2 : {x2, x1} ≻ ∅ Pg : {y} ≻ ∅

Theorem

Acyclicity or transferable utility is necessary for stability!

“Stability and CE in Trading Networks” (Hatfield–K.–Nichifor–Ostrovsky–Westkamp)

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Frontiers of Matching Theory

Conclusion

Acyclicity and substitutability are necessary and sufficient for (classical) matching theory...

...and at the outer frontiers, surprising structure arises.

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Frontiers of Matching Theory

Conclusion

Acyclicity and substitutability are necessary and sufficient for (classical) matching theory...

...and at the outer frontiers, surprising structure arises.

Open Questions

Optimal contract language? Necessary conditions for many-to-one stability? Matching with complementarities?

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Frontiers of Matching Theory

Conclusion

Acyclicity and substitutability are necessary and sufficient for (classical) matching theory...

...and at the outer frontiers, surprising structure arises.

Open Questions

Optimal contract language? Necessary conditions for many-to-one stability? Matching with complementarities?

QED

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Frontiers of Matching Theory Extra Slides

Extra Slides

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SLIDE 97

Frontiers of Matching Theory Extra Slides

Related Literature

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SLIDE 98

Frontiers of Matching Theory Extra Slides

Related Literature

Gale–Shapley (1962)

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SLIDE 99

Frontiers of Matching Theory Extra Slides

Related Literature

Roth (1986) Gale–Shapley (1962)

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SLIDE 100

Frontiers of Matching Theory Extra Slides

Related Literature

Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962)

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SLIDE 101

Frontiers of Matching Theory Extra Slides

Related Literature

Hatfield–K. (2010a) Ostrovsky (2008) Klaus–Walzl (2009) Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962)

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Frontiers of Matching Theory Extra Slides

Related Literature

Hatfield–K. (2010b) Hatfield–K. (2010a) Ostrovsky (2008) Klaus–Walzl (2009) Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962)

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SLIDE 103

Frontiers of Matching Theory Extra Slides

Related Literature

Hatfield–K. (2010b) Hatfield–K. (2010a) Ostrovsky (2008) Klaus–Walzl (2009) Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962)

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SLIDE 104

Frontiers of Matching Theory Extra Slides

Related Literature

Hatfield–K. (2010b) Hatfield–K. (2010a) Ostrovsky (2008) Klaus–Walzl (2009) Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962)

Kara–S¨

  • nmez (1996, 1997), Gul–Stachetti (1999), Haake–Klaus (2008a,b),

Hatfield–Kojima (2009), Jaume et al. (2009), . . .

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Frontiers of Matching Theory Extra Slides

When are preferences substitutable?

Subdividing reveals Substitutability ≻h:

  • xα, zβ

≻

  • xα,β

≻ {xα} ≻

  • zβ

≻

  • xβ

≻′

h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

Subdividing thwarts Substitutability ≻d: {x40} ≻ ∅ ≻′

d: {x20, x20′} ≻ ∅

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y ))

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y ))

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y )) X

  • xβ, x′, z′

{z′}

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y )) X

  • xβ, x′, z′

{z′}

  • xα, xβ, x′, zβ
  • x′, xβ, zβ, z′
  • xβ, z′

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y )) X

  • xβ, x′, z′

{z′}

  • xα, xβ, x′, zβ
  • x′, xβ, zβ, z′
  • xβ, z′
  • xα, x′, zβ
  • xα, xβ, x′, zβ, z′

{xβ, z′}

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SLIDE 112

Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y )) X

  • xβ, x′, z′

{z′}

  • xα, xβ, x′, zβ
  • x′, xβ, zβ, z′
  • xβ, z′
  • xα, x′, zβ
  • xα, xβ, x′, zβ, z′

{xβ, z′}

  • xα, x′, zβ
  • xα, xβ, x′, zβ, z′

{xβ, z′}

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Frontiers of Matching Theory Extra Slides

Substitutability ⇒ Stability

Proof by “Generalized Deferred Acceptance”

Φ(Y ) = X − RH(X − RD(Y )) ≻h:

  • xα, zβ

≻

  • xα, xβ

≻ {xα} ≻

  • zβ

≻

  • xβ

≻h′: {x′} ≻ {z′} ≻xD:

  • xβ, x′

≻ {xα, x′} ≻

  • xβ

≻ {x′} ≻ {xα} ≻zD: {z′} ≻

  • zβ

Y X − RD(Y ) RH(X − RD(Y )) X

  • xβ, x′, z′

{z′}

  • xα, xβ, x′, zβ
  • x′, xβ, zβ, z′
  • xβ, z′
  • xα, x′, zβ
  • xα, xβ, x′, zβ, z′

{xβ, z′}

  • xα, x′, zβ
  • xα, xβ, x′, zβ, z′

{xβ, z′}

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Frontiers of Matching Theory Extra Slides

Full Substitutability ⇒ Guaranteed Stability

Proof by “Generalized Deferred Acceptance”

ΦS

  • X B, X S

:= X − RB

  • X B|X S

ΦB

  • X B, X S

:= X − RS

  • X S|X B

Φ

  • X B, X S

=

  • ΦB
  • X B, X S

, ΦS

  • X B, X S

If X is acyclic, preferences are fully substitutable, and Φ

  • X B, X S

=

  • X B, X S

, then X B ∩ X S stable. If X is acyclic, preferences are fully substitutable, and A is stable, then there exist X B, X S ⊆ X such that Φ

  • X B, X S

=

  • X B, X S

with X B ∩ X S = A. If preferences are fully substitutable, then Φ is isotone.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Definition

A set of contracts

  • x1, . . . , xN

is a chain if

1 xn

B = xn+1 S

for all n = 1, . . . , N − 1.

2 xn

S = xm S implies that n = m.

3 xN

B = x1 S .

Definition (Ostrovsky, 2008)

An allocation A is chain stable if it is individually rational and there is no chain that is a blocking set.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Theorem

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then an allocation is stable if and only if it is chain stable.

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice.

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Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice.

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Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅.

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Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅. ...is strictly weaker than stability when preferences are not fully substitutable.

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Frontiers of Matching Theory Extra Slides

Chain Stability

Corollary

Suppose that the set of contracts X is acyclic and that preferences are fully substitutable. Then, the set of chain stable allocations is a nonempty lattice. But chain stability...

...is unappealing when X is cyclic. F = {f , g}; xS = yB = f ; xB = yS = g; Pf : {x, y} ≻ ∅, Pg : {x, y} ≻ ∅. ...is strictly weaker than stability when preferences are not fully substitutable. ...does not correspond to standard many-to-many stability.

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Frontiers of Matching Theory Extra Slides

The Laws of Aggregate Demand and Supply

Definition

The preferences of f ∈ F satisfy the Law of Aggregate Demand (LoAD) if for all Y , Y ′, Z ⊆ X such that Y ′ ⊆ Y

  • C f

B (Y |Z)

  • −
  • C f

B (Y ′|Z)

  • ≥
  • C f

S (Z|Y )

  • −
  • C f

S (Z|Y ′)

  • .

Definition

The preferences of f ∈ F satisfy the Law of Aggregate Supply (LoAS) if for all Y , Z, Z ′ ⊆ X such that Z ′ ⊆ Z

  • C f

S (Z|Y )

  • −
  • C f

S (Z ′|Y )

  • ≥
  • C f

B (Y |Z)

  • −
  • C f

B (Y |Z ′)

  • .

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Frontiers of Matching Theory

Index

Stable Marriage Deferred Acceptance Stability Lattice Structure Opposition of Interests “Lone Wolf” Theorem Generalizations Matching with Contracts Substitutability = ⇒ Stability Literature Survey Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets Stability ⇐ ⇒ Subs Extensions

LoAD/“Lone Wolf” Language Completion Money

Conclusion Lang/GDA Example Chain Stability LoAD/LoAS

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