What Matchings Can be Stable? Refutability in Matching Theory

Transcription

1 allis/thomson Conference What Matchings Can be Stable? Refutability in Matching Theory Federico Echenique California Institute of Technology April 21-22, 2006

2 Motivation Wallis/Thomson Conference Standard problem in matching theory.

3 allis/thomson Conference Motivation Standard problem in matching theory. Given: agents preferences

4 allis/thomson Conference Motivation Standard problem in matching theory. Given: agents preferences Predict: matchings.

5 Wallis/Thomson Conference Motivation Standard problem in matching theory. Given: agents preferences Predict: matchings. New problem Given: agents matchings µ 1,..., µ K

6 Wallis/Thomson Conference 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?

7 Wallis/Thomson Conference 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?

8 Wallis/Thomson Conference Results vaguely Testing (Two-sided) Matching Theory: Given Observations: agents & matchings (who matches to whom). Unobservables: preferences.

9 Wallis/Thomson Conference 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?

10 Wallis/Thomson Conference 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?

11 Wallis/Thomson Conference 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.

12 Wallis/Thomson Conference 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, Kübler, Bossert & Sprumont, Chappori, Ekeland, Kübler & Polemarchakis. Game Theory: Ledyard, Sprumont, Zhou, Zhou & Ray, Galambos.

13 Wallis/Thomson Conference 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, Kübler, Bossert & Sprumont, Chappori, Ekeland, Kübler & Polemarchakis. Game Theory: Ledyard, Sprumont, Zhou, Zhou & Ray, Galambos. Matching?

14 Motivation II Wallis/Thomson Conference Is this interesting?

15 allis/thomson Conference Motivation II Is this interesting? Matching as a positive theory. many recent empirical papers on matching.

16 Motivation II Wallis/Thomson Conference 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.

17 The Model allis/thomson Conference Two finite, disjoint, sets M (men) and W (women).

18 allis/thomson Conference 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.

19 The Model Preferences Wallis/Thomson Conference A preference relation is a linear, transitive and antisymmetric binary relation. P(m) is over W { } P(w) is over M { }

20 The Model Preferences allis/thomson Conference 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 ).

21 The Model Preferences allis/thomson Conference 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.

22 Stability Definition allis/thomson Conference µ is individually rational if a M W, µ(a) µ(a) P(a).

23 Stability Definition Wallis/Thomson Conference 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).

24 Stability Definition allis/thomson Conference 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.

25 Stability Definition allis/thomson Conference 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.

26 allis/thomson Conference Gale-Shapley (1962) Theorem S(P) is non-empty and a man-best/woman-worst and a woman-best/man-worst matching.

27 Statement of the problem Wallis/Thomson Conference Let H = {µ 1,... µ n } M. Is there a preference profile P such that H S(P)?

28 Statement of the problem allis/thomson Conference 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.

29 Wallis/Thomson Conference Let M = W. µ(a) for all a and all µ H. (this is WLOG)

30 allis/thomson Conference Proposition If M 3, then M is not rationalizable.

31 Proof Wallis/Thomson Conference

32 Proof Wallis/Thomson Conference µ M m 1 w 1 m 2 w 2 m 3 w 3

33 Proof Wallis/Thomson Conference µ M m 1 w 1 m 2 w 2 m 3 w 3 µ W m 1 1 w m 2 2 w m 3 w 3

34 Proof Wallis/Thomson Conference µ M m 1 w 1 m 2 w 2 m 3 w 3 µ W m 1 1 w m 2 2 w m 3 w 3 µ m 1 m 2 m 3 w 1 w 2 w 3

35 Proof Wallis/Thomson Conference µ M m 1 w 1 m 2 w 2 m 3 w 3 µ W m 1 1 w m 2 2 w m 3 w 3 µ m 1 m 2 m 3 w 1 w 2 w 3 ˆµ m 1 w 2 m 3 w 3

36 allis/thomson Conference Proposition If, for all m, µ i (m) µ j (m) for all µ i, µ j H, then H is rationalizable.

37 allis/thomson Conference Proof. m w µ 1 (m) µ n (w) µ 2 (m) µ n 1 (w).. µ n (m) µ 1 (w)

38 allis/thomson Conference So: Matching theory is refutable (everything is not rationalizable)

39 allis/thomson Conference So: Matching theory is refutable (everything is not rationalizable) Source of refutability is: µ(a) = µ (a) for some agents a.

40 Example allis/thomson Conference m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2 µ 3 w 2 w 3 w 1 w 4 Can you find P s.t. µ 1, µ 2 and µ 3 are stable?

41 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

42 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

43 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

44 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

45 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

46 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

47 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

48 Example allis/thomson Conference How do the m compare µ 1 (m) and µ 2 (m)? m 1 m 2 m 3 m 4 µ 1 w 1 w 2 w 3 w 4 µ 2 w 1 w 3 w 4 w 2

49 Wallis/Thomson Conference µ 1 µ 2 : m 1 m 2 m 3 m 4 m 2 µ 1 µ 3 : m 1 m 3 m 4 µ 2 µ 3 : m 1 m 3 m 4 m 2

50 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3.

51 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C.

52 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C. m 2 C, and µ 2 (m 2 ) = µ 3 (m 2 ) µ 3 (m 2 ) P(m 2 ) µ 1 (m 2 ).

53 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C. m 2 C, and µ 2 (m 2 ) = µ 3 (m 2 ) µ 3 (m 2 ) P(m 2 ) µ 1 (m 2 ). µ 3 (m) P(m) µ 1 (m) m C.

54 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C. m 2 C, and µ 2 (m 2 ) = µ 3 (m 2 ) µ 3 (m 2 ) P(m 2 ) µ 1 (m 2 ). µ 3 (m) P(m) µ 1 (m) m C. But, m 4 C and µ 1 (m 4 ) = µ 3 (m 4 ) so µ 2 (m) P(m) µ 3 (m) m C.

55 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C. m 2 C, and µ 2 (m 2 ) = µ 3 (m 2 ) µ 3 (m 2 ) P(m 2 ) µ 1 (m 2 ). µ 3 (m) P(m) µ 1 (m) m C. But, m 4 C and µ 1 (m 4 ) = µ 3 (m 4 ) so µ 2 (m) P(m) µ 3 (m) m C. Now: m 1 C C, so µ 2 (m 1 ) P(m 1 ) µ 3 (m 1 ) P(m 1 ) µ 1 (m 1 ).

56 Wallis/Thomson Conference Example (cont.) all m C = {m 2, m 3, m 4 } agree on µ 1 and µ 2 ; all m C = {m 1, m 2, m 3 } agree on µ 1 and µ 3 ; all m C = {m 1, m 3, m 4 } agree on µ 2 and µ 3. Say µ 2 (m) P(m) µ 1 (m) m C. m 2 C, and µ 2 (m 2 ) = µ 3 (m 2 ) µ 3 (m 2 ) P(m 2 ) µ 1 (m 2 ). µ 3 (m) P(m) µ 1 (m) m C. But, m 4 C and µ 1 (m 4 ) = µ 3 (m 4 ) so µ 2 (m) P(m) µ 3 (m) m C. Now: m 1 C C, so H is not rationalizable. µ 2 (m 1 ) P(m 1 ) µ 3 (m 1 ) P(m 1 ) µ 1 (m 1 ).

57 Pairwise graphs allis/thomson Conference Let µ i and µ j with i < j.

58 Pairwise graphs allis/thomson Conference Let µ i and µ j with i < j. Graph (M, E(µ i, µ j )) with

59 allis/thomson Conference Pairwise graphs Let µ i and µ j with i < j. Graph (M, E(µ i, µ j )) with vertex-set M (m, m ) E(µ i, µ j ) iff µ i (m) = µ j (m ).

60 allis/thomson Conference Pairwise graphs Let µ i and µ j with i < j. Graph (M, E(µ i, µ j )) with vertex-set M (m, m ) E(µ i, µ j ) iff µ i (m) = µ j (m ). Let C(µ i, µ j ) the set of all connected components of (M, E(µ i, µ j )).

61 allis/thomson Conference Female pairwise graphs (W, F (µ i, µ j )) vertex-set W (w, w ) F (µ i, µ j ) if µ j (w) = µ i (w). Lemma The following statements are equivalent: 1. C is a connected component of (M, E(µ i, µ j )) 2. µ i (C) is a connected component of (W, F (µ i, µ j )) In addition, if C is a connected component of (M, E(µ i, µ j )), then µ j (C) = µ i (C).

62 allis/thomson Conference Coincidence/conflict of interest Lemma Let H be rationalized by preference profile P. If µ i, µ j H, and C C(µ i, µ j ), then either (1) or (2) hold. µ i (m) P(m) µ j (m) m C&µ j (w) P(w) µ i (w) w µ i (C) (1) µ j (m) P(m) µ i (m) m C&µ i (w) P(w) µ j (w) w µ i (C) (2)

63 allis/thomson Conference Coincidence/conflict of interest Lemma Let H be rationalized by preference profile P. If µ i, µ j H, and C C(µ i, µ j ), then either (1) or (2) hold. µ i (m) P(m) µ j (m) m C&µ j (w) P(w) µ i (w) w µ i (C) (1) µ j (m) P(m) µ i (m) m C&µ i (w) P(w) µ j (w) w µ i (C) (2) Further, if P is a preference profile such that: for all µ i, µ j H, and C C(µ i, µ j ), either (1) or (2) hold, and in addition then P rationalizes H. P(m) w w / {µ(m) : µ H} P(w) m m / {µ(w) : µ H},

64 allis/thomson Conference Lattice operations. C C(µ i, µ j ), either (3) or (4) must hold: (µ i µ j ) C = µ i C and (µ i µ j ) C = µ j C (3) (µ i µ j ) C = µ j C and (µ i µ j ) C = µ i C. (4)

65 Def. Binary relation allis/thomson Conference Let C ij C(µ i, µ j ) C ik C(µ i, µ k )

66 Def. Binary relation allis/thomson Conference Let C ij C(µ i, µ j ) C ik C(µ i, µ k ) If m C ij C ik with µ j (m) = µ k (m), then say C ij C ik

67 Def. Binary relation allis/thomson Conference Let C ij C(µ i, µ j ) C ik C(µ i, µ k ) If m C ij C ik with µ j (m) = µ k (m), then say C ij C ik ( m C ij ) (µ i ( m)p( m)µ j ( m)) iff ( m C ik ) (µ i ( m)p( m)µ k ( m))

68 Def. Binary relation allis/thomson Conference Let C ij C(µ i, µ j ) C ki C(µ k, µ i ) If m C ij C ki with µ j (m) = µ k (m), then say C ij C ki

69 Def. Binary relation allis/thomson Conference Let C ij C(µ i, µ j ) C ki C(µ k, µ i ) If m C ij C ki with µ j (m) = µ k (m), then say C ij C ki ( m C ij ) (µ i ( m)p( m)µ j ( m)) iff ( m C ki ) (µ i ( m)p( m)µ k ( m))

70 Wallis/Thomson Conference If H is rationalizable, cannot have C C C C C

71 Necessary Condition allis/thomson Conference Theorem If H is rationalizable then (C, E E ) can have no cycle with an odd number of.

72 Necessary Condition allis/thomson Conference Theorem If H is rationalizable then (C, D E ) can have no cycle with an odd number of.

73 allis/thomson Conference 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).

74 Identification allis/thomson Conference U m 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 Π m M U m essentially different preference profiles.

75 Rationalizing Random Matchings allis/thomson Conference Proposition If k is fixed, lim inf n P {H k is rationalizable } e k(k 1)/2

76 Precedent - I allis/thomson Conference Gale-Shapley-Conway: S(P) is a NDL

77 Precedent - I allis/thomson Conference Gale-Shapley-Conway: S(P) is a NDL Blair: Any NDL is isomorphic to the core of some matching market

78 allis/thomson Conference Precedent - II Roth-Sotomayor: We might hope to say something more about what kinds of 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 of investigation will not bear any further fruit.