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What Matchings Can be Stable? Refutability in Matching Theory - - PowerPoint PPT Presentation

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What Matchings Can be Stable? Refutability in Matching Theory Federico Echenique California Institute of Technology April 21-22, 2006 Wallis/Thomson Conference Echenique Matchings that can be stable. Motivation Standard problem in

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What Matchings Can be Stable? Refutability in Matching Theory

Federico Echenique

California Institute of Technology

April 21-22, 2006

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory. Given:

◮ agents ◮ preferences

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory. Given:

◮ agents ◮ preferences

Predict: matchings.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory. Given:

◮ agents ◮ preferences

Predict: matchings. New problem — Given:

◮ agents ◮ matchings µ1, . . . , µK

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory. Given:

◮ agents ◮ preferences

Predict: matchings. New problem — Given:

◮ agents ◮ matchings µ1, . . . , µK

Are there preferences s.t. µ1, . . . , µK are stable ?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation

Standard problem in matching theory. Given:

◮ agents ◮ preferences

Predict: matchings. New problem — Given:

◮ agents ◮ matchings µ1, . . . , µK

Are there preferences s.t. µ1, . . . , µK are stable ? i.e. can you rationalize µ1, . . . , µK using matching theory ?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Results – vaguely

Testing (Two-sided) Matching Theory: Given

◮ Observations: agents & matchings (who matches to whom). ◮ Unobservables: preferences.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Results – vaguely

Testing (Two-sided) Matching Theory: Given

◮ Observations: agents & matchings (who matches to whom). ◮ Unobservables: preferences.

Problems:

◮ Can you test the theory ? ◮ How do you test it ?

i.e. What are its testable implications?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Results – vaguely

Testing (Two-sided) Matching Theory: Given

◮ Observations: agents & matchings (who matches to whom). ◮ Unobservables: preferences.

Problems:

◮ Can you test the theory ? Yes ◮ How do you test it ?

i.e. What are its testable implications?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Results – vaguely

Testing (Two-sided) Matching Theory: Given

◮ Observations: agents & matchings (who matches to whom). ◮ Unobservables: preferences.

Problems:

◮ Can you test the theory ? Yes ◮ How do you test it ?

i.e. What are its testable implications? I find a specific source of test. impl.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Refutability in Economics

◮ Consumer and producer theory: Samuelson, Afriat, Varian,

Diewert, McFadden, Hanoch & Rothschild, Richter, Matzkin & Richter.

◮ General Equilibrium Theory: Sonnenschein, Mantel, Debreu,

Mas-Colell, Brown & Matzkin, Brown & Shannon, K¨ ubler, Bossert & Sprumont, Chappori, Ekeland, K¨ ubler & Polemarchakis.

◮ Game Theory: Ledyard, Sprumont, Zhou, Zhou & Ray,

Galambos.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Refutability in Economics

◮ Consumer and producer theory: Samuelson, Afriat, Varian,

Diewert, McFadden, Hanoch & Rothschild, Richter, Matzkin & Richter.

◮ General Equilibrium Theory: Sonnenschein, Mantel, Debreu,

Mas-Colell, Brown & Matzkin, Brown & Shannon, K¨ ubler, Bossert & Sprumont, Chappori, Ekeland, K¨ ubler & Polemarchakis.

◮ Game Theory: Ledyard, Sprumont, Zhou, Zhou & Ray,

Galambos.

◮ Matching ?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation – II

Is this interesting?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation – II

Is this interesting?

◮ Matching as a positive theory.

many recent empirical papers on matching.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Motivation – II

Is this interesting?

◮ Matching as a positive theory.

many recent empirical papers on matching.

◮ Applications:

◮ Marriages of “types.” ◮ Hospital-interns matches outside the NRMP. ◮ Student-schools outside of NY. Wallis/Thomson Conference Echenique – Matchings that can be stable.

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The Model

Two finite, disjoint, sets M (men) and W (women).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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The Model

Two finite, disjoint, sets M (men) and W (women). A matching is a function µ : M ∪ W → M ∪ W ∪ {∅} s.t.

  • 1. µ (w) ∈ M ∪ {∅},
  • 2. µ (m) ∈ W ∪ {∅},
  • 3. and m = µ (w) iff w = µ (m).

Denote the set of all matchings by M.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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The Model – Preferences

A preference relation is a linear, transitive and antisymmetric binary relation. P(m) is over W ∪ {∅} P(w) is over M ∪ {∅}

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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The Model – Preferences

A preference relation is a linear, transitive and antisymmetric binary relation. P(m) is over W ∪ {∅} P(w) is over M ∪ {∅} A preference profile is a list P of preference relations for men and women, so P =

  • (P(m))m∈M , (P(w))w∈W
  • .

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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The Model – Preferences

A preference relation is a linear, transitive and antisymmetric binary relation. P(m) is over W ∪ {∅} P(w) is over M ∪ {∅} A preference profile is a list P of preference relations for men and women, so P =

  • (P(m))m∈M , (P(w))w∈W
  • .

Note that preferences are strict.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Stability – Definition

µ is individually rational if ∀a ∈ M ∪ W , µ(a) = ∅ ⇒ µ(a) P(a) ∅.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Stability – Definition

µ is individually rational if ∀a ∈ M ∪ W , µ(a) = ∅ ⇒ µ(a) P(a) ∅. A pair (m, w) blocks µ if w = µ(m) and w P(m) µ(m) and m P(w) µ(w).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Stability – Definition

µ is individually rational if ∀a ∈ M ∪ W , µ(a) = ∅ ⇒ µ(a) P(a) ∅. A pair (m, w) blocks µ if w = µ(m) and w P(m) µ(m) and m P(w) µ(w). µ is stable if it is individually rational and there is no pair that blocks µ.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Stability – Definition

µ is individually rational if ∀a ∈ M ∪ W , µ(a) = ∅ ⇒ µ(a) P(a) ∅. A pair (m, w) blocks µ if w = µ(m) and w P(m) µ(m) and m P(w) µ(w). µ is stable if it is individually rational and there is no pair that blocks µ. S(P) is the set of stable matchings.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Gale-Shapley (1962)

Theorem

S(P) is non-empty and ∃ a man-best/woman-worst and a woman-best/man-worst matching.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Statement of the problem

Let H = {µ1, . . . µn} ⊆ M. Is there a preference profile P such that H ⊆ S(P)?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Statement of the problem

Let H = {µ1, . . . µn} ⊆ M. Is there a preference profile P such that H ⊆ S(P)? Say that H can be rationalized if there is such P.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Let |M| = |W |. µ(a) = ∅ for all a and all µ ∈ H. (this is WLOG)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proposition

If |M| ≥ 3, then M is not rationalizable.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof

µM m1

w1

m2

w2

m3

w3

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof

µM m1

w1

m2

w2

m3

w3

µW m1

  • w1

m2

  • w2

m3

  • w3

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof

µM m1

w1

m2

w2

m3

w3

µW m1

  • w1

m2

  • w2

m3

  • w3

µ m1

  • w1

m2

  • w2

m3

  • w3

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof

µM m1

w1

m2

w2

m3

w3

µW m1

  • w1

m2

  • w2

m3

  • w3

µ m1

  • w1

m2

  • w2

m3

  • w3

ˆ µ m1

w2

m3

w3

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proposition

If, for all m, µi(m) = µj(m) for all µi, µj ∈ H, then H is rationalizable.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Proof.

m w µ1(m) µn(w) µ2(m) µn−1(w) . . . . . . µn(m) µ1(w)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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So:

◮ Matching theory is refutable (everything is not rationalizable)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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So:

◮ Matching theory is refutable (everything is not rationalizable) ◮ Source of refutability is: µ(a) = µ′(a) for some agents a.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

m1 m2 m3 m4 µ1 w1 w2 w3 w4 µ2 w1 w3 w4 w2 µ3 w2 w3 w1 w4 Can you find P s.t. µ1, µ2 and µ3 are stable ?

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2 w3 w4 µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∧

w3 w4 µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∧

w3

∨

w4 µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∧

w3

∧

w4 µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∧

w3

∧

w4

∧

µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∨

w3 w4 µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∨

w3 w4

∨

µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example

How do the m compare µ1(m) and µ2(m) ? m1 m2 m3 m4 µ1 w1 w2

∨

w3

∨

w4

∨

µ2 w1 w3 w4 w2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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µ1 − µ2 : m1

  • m2
  • m3
  • m4
  • µ1 − µ3 :

m1

  • m2
  • m3
  • m4
  • µ2 − µ3 :

m1

  • m2
  • m3

m4

  • Wallis/Thomson Conference

Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C. m2 ∈ C, and µ2(m2) = µ3(m2) ⇒ µ3(m2) P(m2) µ1(m2).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C. m2 ∈ C, and µ2(m2) = µ3(m2) ⇒ µ3(m2) P(m2) µ1(m2). ⇒ µ3(m) P(m) µ1(m) ∀m ∈ C ′.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C. m2 ∈ C, and µ2(m2) = µ3(m2) ⇒ µ3(m2) P(m2) µ1(m2). ⇒ µ3(m) P(m) µ1(m) ∀m ∈ C ′. But, m4 ∈ C and µ1(m4) = µ3(m4) so µ2(m) P(m) µ3(m) ∀m ∈ C ′′.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C. m2 ∈ C, and µ2(m2) = µ3(m2) ⇒ µ3(m2) P(m2) µ1(m2). ⇒ µ3(m) P(m) µ1(m) ∀m ∈ C ′. But, m4 ∈ C and µ1(m4) = µ3(m4) so µ2(m) P(m) µ3(m) ∀m ∈ C ′′. Now: m1 ∈ C ′ ∩ C ′′, so µ2(m1) P(m1) µ3(m1) P(m1) µ1(m1).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Example (cont.)

all m ∈ C = {m2, m3, m4} agree on µ1 and µ2; all m ∈ C ′ = {m1, m2, m3} agree on µ1 and µ3; all m ∈ C ′′ = {m1, m3, m4} agree on µ2 and µ3. Say µ2(m) P(m) µ1(m) ∀m ∈ C. m2 ∈ C, and µ2(m2) = µ3(m2) ⇒ µ3(m2) P(m2) µ1(m2). ⇒ µ3(m) P(m) µ1(m) ∀m ∈ C ′. But, m4 ∈ C and µ1(m4) = µ3(m4) so µ2(m) P(m) µ3(m) ∀m ∈ C ′′. Now: m1 ∈ C ′ ∩ C ′′, so µ2(m1) P(m1) µ3(m1) P(m1) µ1(m1). H is not rationalizable.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Pairwise graphs

Let µi and µj with i < j.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Pairwise graphs

Let µi and µj with i < j. Graph (M, E(µi, µj)) with

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Pairwise graphs

Let µi and µj with i < j. Graph (M, E(µi, µj)) with

◮ vertex-set M ◮ (m, m′) ∈ E(µi, µj) iff µi(m) = µj(m′).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Pairwise graphs

Let µi and µj with i < j. Graph (M, E(µi, µj)) with

◮ vertex-set M ◮ (m, m′) ∈ E(µi, µj) iff µi(m) = µj(m′).

Let C(µi, µj) the set of all connected components of (M, E(µi, µj)).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Female pairwise graphs

(W , F(µi, µj))

◮ vertex-set W ◮ (w, w′) ∈ F(µi, µj) if µj(w) = µi(w).

Lemma

The following statements are equivalent:

  • 1. C is a connected component of (M, E(µi, µj))
  • 2. µi(C) is a connected component of (W , F(µi, µj))

In addition, if C is a connected component of (M, E(µi, µj)), then µj(C) = µi(C).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Coincidence/conflict of interest

Lemma

Let H be rationalized by preference profile P. If µi, µj ∈ H, and C ∈ C(µi, µj), then either (1) or (2) hold. µi(m) P(m) µj(m)∀m ∈ C&µj(w) P(w) µi(w)∀w ∈ µi(C) (1) µj(m) P(m) µi(m)∀m ∈ C&µi(w) P(w) µj(w)∀w ∈ µi(C) (2)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 63

Coincidence/conflict of interest

Lemma

Let H be rationalized by preference profile P. If µi, µj ∈ H, and C ∈ C(µi, µj), then either (1) or (2) hold. µi(m) P(m) µj(m)∀m ∈ C&µj(w) P(w) µi(w)∀w ∈ µi(C) (1) µj(m) P(m) µi(m)∀m ∈ C&µi(w) P(w) µj(w)∀w ∈ µi(C) (2) Further, if P is a preference profile such that: for all µi, µj ∈ H, and C ∈ C(µi, µj), either (1) or (2) hold, and in addition ∅ P(m) w ⇔ w / ∈ {µ(m) : µ ∈ H} ∅ P(w) m ⇔ m / ∈ {µ(w) : µ ∈ H} , then P rationalizes H.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Lattice operations.

C ∈ C(µi, µj), either (3) or (4) must hold: (µi ∧ µj)|C = µi|C and (µi ∨ µj)|C = µj|C (3) (µi ∧ µj)|C = µj|C and (µi ∨ µj)|C = µi|C. (4)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 65
  • Def. Binary relation △

Let Cij ∈ C(µi, µj) Cik ∈ C(µi, µk)

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 66
  • Def. Binary relation △

Let Cij ∈ C(µi, µj) Cik ∈ C(µi, µk) If ∃m ∈ Cij ∩ Cik with µj(m) = µk(m), then say Cij △ Cik

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 67
  • Def. Binary relation △

Let Cij ∈ C(µi, µj) Cik ∈ C(µi, µk) If ∃m ∈ Cij ∩ Cik with µj(m) = µk(m), then say Cij △ Cik (∀ ˜ m ∈ Cij) (µi( ˜ m)P( ˜ m)µj( ˜ m)) iff (∀ ˜ m ∈ Cik) (µi( ˜ m)P( ˜ m)µk( ˜ m))

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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  • Def. Binary relation ▽

Let Cij ∈ C(µi, µj) Cki ∈ C(µk, µi) If ∃m ∈ Cij ∩ Cki with µj(m) = µk(m), then say Cij ▽ Cki

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 69
  • Def. Binary relation ▽

Let Cij ∈ C(µi, µj) Cki ∈ C(µk, µi) If ∃m ∈ Cij ∩ Cki with µj(m) = µk(m), then say Cij ▽ Cki (∀ ˜ m ∈ Cij) (µi( ˜ m)P( ˜ m)µj( ˜ m)) iff (∀ ˜ m ∈ Cki) (µi( ˜ m)P( ˜ m)µk( ˜ m))

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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If H is rationalizable, cannot have C △ C ′ ▽ C ′′ △ C ′′′ △ C

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Necessary Condition

Theorem

If H is rationalizable then (C, E△ ∪ E▽) can have no cycle with an

  • dd number of ▽.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 72

Necessary Condition

Theorem

If H is rationalizable then (C, D△ ∪ E▽) can have no cycle with an

  • dd number of ▽.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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Necessary and Sufficient Condition

Theorem

H is rationalizable if and only if (C, D△ ∪ E▽) has no cycle with an odd number of ▽s, and for the resulting graph (C, D), there is a function d : C → {−1, 1} that satisfies:

  • 1. C ▽ C′ ⇒ d(C) + d(C′) = 0,
  • 2. (C, C′, C′′) ∈ B ⇒ (d(C) + d(C′)) d(C′′) ≥ 0.

Further, there is a rationalizing preference profile for each function d satisfying (1) and (2).

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 74

Identification

Um is the set of women m is not matched to in any µ ∈ H.

Proposition

If H is rationalizable, then it is rationalizable by at least (2 |M|)|M| Πm∈M |Um| essentially different preference profiles.

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 75

Rationalizing Random Matchings

Proposition

If k is fixed, lim inf

n→∞ P {Hk is rationalizable } ≥ e−k(k−1)/2

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 76

Precedent - I

Gale-Shapley-Conway: S(P) is a NDL

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 77

Precedent - I

Gale-Shapley-Conway: S(P) is a NDL Blair: Any NDL is isomorphic to the core of some matching market

Wallis/Thomson Conference Echenique – Matchings that can be stable.

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SLIDE 78

Precedent - II

Roth-Sotomayor: We might hope to say something more about what kinds

  • f lattices arise as sets of stable matchings, in order to

use any additional properties thus specified to learn more about the market. (Blair’s) Theorem shows that this line

  • f investigation will not bear any further fruit.

Wallis/Thomson Conference Echenique – Matchings that can be stable.