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

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

Motivation Standard problem in matching theory. Given: ◮ agents ◮ preferences Wallis/Thomson Conference Echenique – Matchings that can be stable.

Motivation Standard problem in matching theory. Given: ◮ agents ◮ preferences Predict: matchings. Wallis/Thomson Conference Echenique – Matchings that can be stable.

Motivation Standard problem in matching theory. Given: ◮ agents ◮ preferences Predict: matchings. New problem — Given: ◮ agents ◮ matchings µ 1 , . . . , µ K Wallis/Thomson Conference Echenique – Matchings that can be stable.

Motivation Standard problem in matching theory. Given: ◮ agents ◮ preferences Predict: matchings. New problem — Given: ◮ agents ◮ matchings µ 1 , . . . , µ K Are there preferences s.t. µ 1 , . . . , µ K are stable ? Wallis/Thomson Conference Echenique – Matchings that can be stable.

Motivation Standard problem in matching theory. Given: ◮ agents ◮ preferences Predict: matchings. New problem — Given: ◮ agents ◮ matchings µ 1 , . . . , µ K Are there preferences s.t. µ 1 , . . . , µ K are stable ? i.e. can you rationalize µ 1 , . . . , µ K using matching theory ? Wallis/Thomson Conference Echenique – Matchings that can be stable.

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.

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.

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.

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.

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.

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.

Motivation – II Is this interesting? Wallis/Thomson Conference Echenique – Matchings that can be stable.

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.

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.

The Model Two finite, disjoint, sets M (men) and W (women). Wallis/Thomson Conference Echenique – Matchings that can be stable.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Statement of the problem Let H = { µ 1 , . . . µ n } ⊆ M . Is there a preference profile P such that H ⊆ S ( P )? Wallis/Thomson Conference Echenique – Matchings that can be stable.

Statement of the problem Let H = { µ 1 , . . . µ n } ⊆ 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.

Let | M | = | W | . µ ( a ) � = ∅ for all a and all µ ∈ H . (this is WLOG) Wallis/Thomson Conference Echenique – Matchings that can be stable.

Proposition If | M | ≥ 3 , then M is not rationalizable. Wallis/Thomson Conference Echenique – Matchings that can be stable.

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

Proof µ M � w 1 m 1 � � w 2 m 2 � � w 3 m 3 � Wallis/Thomson Conference Echenique – Matchings that can be stable.

� � � � Proof µ M µ W � w 1 m 1 � m 1 � w 1 � � � � � � � � � � � � � � � w 2 m 2 � m 2 � � w 2 � � � � � � � � � � � � � � � � � w 3 m 3 � m 3 w 3 Wallis/Thomson Conference Echenique – Matchings that can be stable.

� � � � � � � � � � Proof µ M µ W µ � w 1 m 1 � m 1 � w 1 m 1 w 1 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � w 2 � m 2 � m 2 � � w 2 m 2 w 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � w 3 m 3 � m 3 w 3 m 3 w 3 Wallis/Thomson Conference Echenique – Matchings that can be stable.