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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

  1. Generalized Matching Theory Matching with Contracts Matching with Contracts Hatfield & Kominers June 4, 2012 9

  2. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over contracts involving him, Hatfield & Kominers June 4, 2012 9

  3. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over 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

  4. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over 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

  5. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over 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. x D identifies the doctor of contract x ; Hatfield & Kominers June 4, 2012 9

  6. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over 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. x D identifies the doctor of contract x ; x H identifies the hospital of contract x . Hatfield & Kominers June 4, 2012 9

  7. Generalized Matching Theory Matching with Contracts Matching with Contracts A set of doctors D : each doctor has a strict preference order over 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. x D identifies the doctor of contract x ; x H identifies the hospital of contract x . An outcome is a set of contracts Y ⊆ X such that if x , z ∈ Y and x D = z D , then x = z . Hatfield & Kominers June 4, 2012 9

  8. Generalized Matching Theory Matching with Contracts Choice Functions Hatfield & Kominers June 4, 2012 10

  9. Generalized Matching Theory Matching with Contracts Choice Functions C d ( Y ) ≡ max P d { x ∈ Y : x D = d } . Hatfield & Kominers June 4, 2012 10

  10. Generalized Matching Theory Matching with Contracts Choice Functions C d ( Y ) ≡ max P d { x ∈ Y : x D = d } . C h ( Y ) ≡ max P h { Z ⊆ Y : Z H = { h }} . Hatfield & Kominers June 4, 2012 10

  11. Generalized Matching Theory Matching with Contracts Choice Functions C d ( Y ) ≡ max P d { x ∈ Y : x D = d } . C h ( Y ) ≡ max P h { Z ⊆ Y : Z H = { h }} . We define the rejection functions R D ( Y ) ≡ Y − ∪ d ∈ D C d ( Y ) , R H ( Y ) ≡ Y − ∪ h ∈ H C h ( Y ) . Hatfield & Kominers June 4, 2012 10

  12. Generalized Matching Theory Matching with Contracts Choice Functions C d ( Y ) ≡ max P d { x ∈ Y : x D = d } . C h ( Y ) ≡ max P h { Z ⊆ Y : Z H = { h }} . We define the rejection functions R D ( Y ) ≡ Y − ∪ d ∈ D C d ( Y ) , R H ( Y ) ≡ Y − ∪ h ∈ H C h ( Y ) . Definition The preferences of hospital h are substitutable if for all Y ⊆ X , if ∈ C h ( Y ), then z / ∈ C h ( { x } ∪ Y ) for all x � = z . z / Hatfield & Kominers June 4, 2012 10

  13. 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 x D = d , then x ≻ d ∅ , for all h ∈ H , C h ( A ) = A H . 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

  14. 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 x D = d , then x ≻ d ∅ , for all h ∈ H , C h ( A ) = A H . 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

  15. Generalized Matching Theory Matching with Contracts Characterization of Stable Outcomes Consider the operator � X D � � X D � Φ H ≡ X − R D � X H � � X H � Φ D ≡ X − R H X D , X H � X H � X D �� � � � � Φ = Φ D , Φ H Hatfield & Kominers June 4, 2012 12

  16. Generalized Matching Theory Matching with Contracts Characterization of Stable Outcomes Consider the operator � X D � � X D � Φ H ≡ X − R D � X H � � X H � Φ D ≡ X − R H X D , X H � X H � X D �� � � � � Φ = Φ D , Φ H Theorem Suppose that the preferences of hospitals are substitutable. Then if , the outcome X D ∩ X H is stable. � X D , X H � � X D , X H � Φ = Conversely, if A is a stable outcome, there exist X D , X H ⊆ X such and X D ∩ X H = A. � X D , X H � � X D , X H � that Φ = Hatfield & Kominers June 4, 2012 12

  17. Generalized Matching Theory Matching with Contracts Existence of Stable Allocations Theorem Suppose that hospitals’ preferences are substitutable. Then there � X D , X H � exists a nonempty finite lattice of fixed points of Φ which correspond to stable outcomes A = X D ∩ X H . Hatfield & Kominers June 4, 2012 13

  18. Generalized Matching Theory Matching with Contracts Existence of Stable Allocations Theorem Suppose that hospitals’ preferences are substitutable. Then there � X D , X H � exists a nonempty finite lattice of fixed points of 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

  19. Generalized Matching Theory Matching with Contracts Existence of Stable Allocations Theorem Suppose that hospitals’ preferences are substitutable. Then there � X D , X H � exists a nonempty finite lattice of fixed points of 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Generalized Matching Theory Matching with Contracts Matching Without Substitutes Hatfield & Kominers June 4, 2012 16

  26. Generalized Matching Theory Matching with Contracts Matching Without Substitutes Substitutability is sufficient, but is it “necessary”? Hatfield & Kominers June 4, 2012 16

  27. 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

  28. 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

  29. 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¨ onmez and Switzer (2011), S¨ onmez (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

  30. 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¨ onmez and Switzer (2011), S¨ onmez (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

  31. � � Generalized Matching Theory Supply Chain Matching Supply Chain Matching (Ostrovsky, 2008) s Same-side contracts are substitutes . Cross-side contracts are i � � � complements . � � � � � � � � � � � � � � ⇒ Objects are b 1 b 2 fully substitutable . Hatfield & Kominers June 4, 2012 17

  32. � � Generalized Matching Theory Supply Chain Matching Supply Chain Matching (Ostrovsky, 2008) s Same-side contracts are substitutes . Cross-side contracts are i � � � complements . � � � � � � � � � � � � � � ⇒ Objects are b 1 b 2 fully substitutable . Theorem Stable outcomes exist. Hatfield & Kominers June 4, 2012 17

  33. 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

  34. 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

  35. 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

  36. � � Generalized Matching Theory Supply Chain Matching Cyclic Contract Sets g � ������� P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ y P f 2 : { x 2 , x 1 } ≻ ∅ f 1 x 1 x 2 P g : { y } ≻ ∅ f 2 Hatfield & Kominers June 4, 2012 19

  37. � � Generalized Matching Theory Supply Chain Matching Cyclic Contract Sets g � ������� P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ y P f 2 : { x 2 , x 1 } ≻ ∅ f 1 x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity is necessary for stability. Hatfield & Kominers June 4, 2012 19

  38. 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

  39. � � � 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 P s : { x } ≻ { z } ≻ ∅ x P i : { x , y } ≻ ∅ i z y P b : { z } ≻ { x } ≻ ∅ b Hatfield & Kominers June 4, 2012 20

  40. 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

  41. Generalized Matching Theory The Assignment Problem The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972) ζ m , w ∼ total surplus of marriage of man m and woman w Hatfield & Kominers June 4, 2012 21

  42. Generalized Matching Theory The Assignment Problem The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972) ζ m , w ∼ total surplus of marriage of man m and woman w � 1 m , w married assignment indicators : a m , w ≡ 0 otherwise Hatfield & Kominers June 4, 2012 21

  43. Generalized Matching Theory The Assignment Problem The Model (Koopmans–Beckmann, 1957; Gale, 1960; Shapley–Shubik, 1972) ζ m , w ∼ total surplus of marriage of man m and woman w � 1 m , w married assignment indicators : a m , w ≡ 0 otherwise Stable assignment (˜ a m , w ) solves the integer program � 0 ≤ � w a m , w ≤ 1 ∀ m � � max a m , w ζ m , w � � 0 ≤ � m a m , w ≤ 1 ∀ w � m w Hatfield & Kominers June 4, 2012 21

  44. Generalized Matching Theory The Assignment Problem “Efficient Mating” z m , w ≡ ζ m , w − ζ m , ∅ − ζ ∅ , w ∼ marital surplus �� � � � � � � max a m , w ζ m , w = max a m , w z m , w + ζ m , ∅ + ζ ∅ , w m w m w m w Theorem Stable assignment maximizes aggregate marriage output. Note Even with a m , w ∈ [0 , 1] , the optimum is always an integer solution. Hatfield & Kominers June 4, 2012 22

  45. 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 ζ m , w = h ( x m , y w ), 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

  46. 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

  47. 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

  48. � � Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks Cyclic Contract Sets g � ������� P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ y P f 2 : { x 2 , x 1 } ≻ ∅ f 1 x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity is necessary for stability! Hatfield & Kominers June 4, 2012 25

  49. 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

  50. 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

  51. 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

  52. 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, p ω ) 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

  53. Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks The Setting: Demand Each agent i has quasilinear utility over arrangements: � � U i ([Ψ; p ]) = u i (Ψ i ) + p ψ − p ψ . ψ ∈ Ψ i → ψ ∈ Ψ → i U i extends naturally to (feasible) outcomes. Hatfield & Kominers June 4, 2012 28

  54. Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks The Setting: Demand Each agent i has quasilinear utility over arrangements: � � U i ([Ψ; p ]) = u i (Ψ i ) + p ψ − p ψ . ψ ∈ Ψ i → ψ ∈ Ψ → i U i extends naturally to (feasible) outcomes. For any price vector p ∈ R | Ω | , the demand of i is D i ( p ) = argmax Ψ ⊆ Ω i U i ([Ψ; p ]) . For any set of contracts Y ⊆ X , the choice of i is C i ( Y ) = argmax Z ⊆ Y i U i ( Z ) . Hatfield & Kominers June 4, 2012 28

  55. Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks Assumptions on Preferences 1 u i (Ψ) ∈ R ∪ {−∞} . Hatfield & Kominers June 4, 2012 29

  56. Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks Assumptions on Preferences 1 u i (Ψ) ∈ R ∪ {−∞} . 2 u i ( ∅ ) ∈ R . Hatfield & Kominers June 4, 2012 29

  57. Generalized Matching Theory Stability and Competitive Equilibrium in Trading Networks Assumptions on Preferences 1 u i (Ψ) ∈ R ∪ {−∞} . 2 u i ( ∅ ) ∈ R . 3 Full substitutability ... Hatfield & Kominers June 4, 2012 29

  58. 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

  59. 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 ⊆ X i such that | C i ( Z ) | = | C i ( Y ) | = 1, 1 if Y i → = Z i → , and Y → i ⊆ Z → i , then for Y ∗ ∈ C i ( Y ) and Z ∗ ∈ C i ( Z ), we have ( Y → i − Y ∗ → i ) ⊆ ( Z → i − Z ∗ → i ) and Y ∗ i → ⊆ Z ∗ i → ; 2 if Y → i = Z → i , and Y i → ⊆ Z i → , then for Y ∗ ∈ C i ( Y ) and Z ∗ ∈ C i ( Z ), we have ( Y i → − Y ∗ i → ) ⊆ ( Z i → − Z ∗ i → ) and Y ∗ → i ⊆ Z ∗ → i . Hatfield & Kominers June 4, 2012 30

  60. 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 V i ( p ) := max Ψ ⊆ Ω i U i ([Ψ; p ]) is submodular (V i ( p ∨ q ) + V i ( p ∧ q ) ≤ V i ( p ) + V i ( q ) ). Hatfield & Kominers June 4, 2012 31

  61. 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 , A i ∈ C i ( A ); 2 Unblocked : There is no nonempty, feasible Z ⊆ X such that Z ∩ A = ∅ and for each i , and for each Y i ∈ C i ( Z ∪ A ), we have Z i ⊆ Y i . Definition Arrangement [Ψ; p ] is a competitive equilibrium (CE) if for each i , Ψ i ∈ D i ( p ) . Hatfield & Kominers June 4, 2012 32

  62. 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 u i to ˆ u i . 2 Associate : Construct a two-sided one-to-many matching market:  i → “firm”: valuation ˜ u i (Ψ) := ˆ u i (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

  63. 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

  64. 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

  65. 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

  66. 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, p ψ ) } is stable. The reverse implication is not true in general. Hatfield & Kominers June 4, 2012 35

  67. 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, p ψ ) } 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, ¯ p ω ) ∈ A, then p ω = ¯ p ω . Hatfield & Kominers June 4, 2012 35

  68. 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

  69. 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

  70. 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 U i ( Z ) > U i ( A ) for all i associated with Z . Hatfield & Kominers June 4, 2012 37

  71. 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 U i ( Z ) > U i ( A ) for all i associated with Z . Definition A set of contracts Z is a chain if its elements can be arranged in some order y 1 , . . . , y | Z | such that s ( y ℓ +1 ) = b ( y ℓ ) for all ℓ < | Z | . Hatfield & Kominers June 4, 2012 37

  72. 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 U i ( Z ) > U i ( A ) for all i associated with Z . Definition A set of contracts Z is a chain if its elements can be arranged in some order y 1 , . . . , y | Z | such that s ( y ℓ +1 ) = b ( y ℓ ) 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 Y i ∈ C i ( Z ∪ A ), we have Z i ⊆ Y i . Hatfield & Kominers June 4, 2012 37

  73. 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 U i ( Z ) > U i ( A ) for all i associated with Z . Definition A set of contracts Z is a chain if its elements can be arranged in some order y 1 , . . . , y | Z | such that s ( y ℓ +1 ) = b ( y ℓ ) 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 Y i ∈ C i ( Z ∪ A ), we have Z i ⊆ Y i . Hatfield & Kominers June 4, 2012 37

  74. 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 U i ( Z ) > U i ( A ) for all i associated with Z . Definition A set of contracts Z is a chain if its elements can be arranged in some order y 1 , . . . , y | Z | such that s ( y ℓ +1 ) = b ( y ℓ ) 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 Z i ⊆ Y i and U i ( Y i ) > U i ( A ). Hatfield & Kominers June 4, 2012 37

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