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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

  1. Generalized Matching Theory Matching with Contracts Matching with Contracts (Hatfield–Milgrom, 2005) x = (doctor , hospital , terms) Scott Duke Kominers (Harvard) October 12, 2010 9

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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 ∈ C f ( Y ∪ { z } ) z ∈ C f ( Y ∪ { x , z } ) . z / but Scott Duke Kominers (Harvard) October 12, 2010 10

  7. 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 ∈ C f ( Y ∪ { z } ) z ∈ C f ( Y ∪ { x , z } ) . z / but Intuition Receiving new offers makes f (weakly) less interested in old offers. Scott Duke Kominers (Harvard) October 12, 2010 10

  8. 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 ∈ C f ( Y ∪ { z } ) z ∈ C f ( Y ∪ { x , z } ) . z / but Intuition Receiving new offers makes f (weakly) less interested in old offers. Equivalent Definition The rejection function R f ( X ′ ) = X ′ − C f ( X ′ ) is monotone. Scott Duke Kominers (Harvard) October 12, 2010 10

  9. 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

  10. 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 − R H ( X − R D ( Y )) Scott Duke Kominers (Harvard) October 12, 2010 11

  11. 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 − R H ( X − R D ( Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D ( Y ). Scott Duke Kominers (Harvard) October 12, 2010 11

  12. 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 − R H ( X − R D ( Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D ( Y ). If R H and R D are monotone, then Φ is monotone. Scott Duke Kominers (Harvard) October 12, 2010 11

  13. 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 − R H ( X − R D ( Y )) Correspondence between fixed points Y of Φ and stable allocations A = C D ( Y ). If R H and R D 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

  14. 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

  15. 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

  16. 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

  17. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  18. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  19. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  20. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  21. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  22. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  23. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  24. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  25. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Scott Duke Kominers (Harvard) October 12, 2010 14

  26. � � Frontiers of Matching Theory Matching in Networks Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity is necessary for stability! Scott Duke Kominers (Harvard) October 12, 2010 14

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. Frontiers of Matching Theory Extensions Scott Duke Kominers (Harvard) October 12, 2010 19

  36. Frontiers of Matching Theory Extensions Surprising generalization of “Lone Wolf” Theorem Agents’ excess stocks are invariant Scott Duke Kominers (Harvard) October 12, 2010 19

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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 � � � C f � � � C f � � � C f � B ( Y | Z ) B ( Y ′ | Z ) S ( Z | Y ) 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

  42. Frontiers of Matching Theory The (Generalized) “Lone Wolf” Theorem Scott Duke Kominers (Harvard) October 12, 2010 21

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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.) Scott Duke Kominers (Harvard) October 12, 2010 21

  49. Frontiers of Matching Theory Bundling of Contract Terms 1 Work and wages contracted simultaneously: Employee Preferences: { x w , $ } ≻ ∅ Employer Preferences: { x w , $ } ≻ ∅ Scott Duke Kominers (Harvard) October 12, 2010 22

  50. Frontiers of Matching Theory Bundling of Contract Terms 1 Work and wages contracted simultaneously: Employee Preferences: { x w , $ } ≻ ∅ Employer Preferences: { x w , $ } ≻ ∅ 2 Work and wages contracted separately: Employee Preferences: { x $ } ≻ { x w , x $ } ≻ ∅ Employer Preferences: { x w } ≻ { x w , x $ } ≻ ∅ Scott Duke Kominers (Harvard) October 12, 2010 22

  51. Frontiers of Matching Theory Bundling of Contract Terms 1 Work and wages contracted simultaneously: Employee Preferences: { x w , $ } ≻ ∅ Employer Preferences: { x w , $ } ≻ ∅ 2 Work and wages contracted separately: Employee Preferences: { x $ } ≻ { x w , x $ } ≻ ∅ Employer Preferences: { x w } ≻ { x w , x $ } ≻ ∅ Scott Duke Kominers (Harvard) October 12, 2010 22

  52. Frontiers of Matching Theory Bundling of Contract Terms 1 Work and wages contracted simultaneously: Employee Preferences: { x w , $ } ≻ ∅ Employer Preferences: { x w , $ } ≻ ∅ 2 Work and wages contracted separately: Employee Preferences: { x $ } ≻ { x w , x $ } ≻ ∅ Employer Preferences: { x w } ≻ { x w , x $ } ≻ ∅ Scott Duke Kominers (Harvard) October 12, 2010 22

  53. Frontiers of Matching Theory Bundling of Contract Terms 1 Work and wages contracted simultaneously: Employee Preferences: { x w , $ } ≻ ∅ Employer Preferences: { x w , $ } ≻ ∅ 2 Work and wages contracted separately: Employee Preferences: { x $ } ≻ { x w , x $ } ≻ ∅ Employer Preferences: { x w } ≻ { x w , x $ } ≻ ∅ “Contract Design and Stability in Matching Markets” (Hatfield–K.) Scott Duke Kominers (Harvard) October 12, 2010 22

  54. Frontiers of Matching Theory Completion of Preferences Consider the case of one hospital h with preferences � x α , z β � � x β � ≻ { x α } ≻ � z β � ≻ h : ≻ , which are not substitutable. Scott Duke Kominers (Harvard) October 12, 2010 23

  55. Frontiers of Matching Theory Completion of Preferences Consider the case of one hospital h with preferences � x α , z β � � x β � ≻ { x α } ≻ � z β � ≻ h : ≻ , which are not substitutable. This hospital h actually has preferences x α , x β � x α , z β � x β � ≻ { x α } ≻ z β � ≻ h : � ≻ � ≻ � � , which ARE substitutable. Scott Duke Kominers (Harvard) October 12, 2010 23

  56. Frontiers of Matching Theory Completion of Preferences Consider the case of one hospital h with preferences � x α , z β � � x β � ≻ { x α } ≻ � z β � ≻ h : ≻ , which are not substitutable. This hospital h actually has preferences x α , x β � x α , z β � x β � ≻ { x α } ≻ z β � ≻ h : � ≻ � ≻ � � , which ARE substitutable. “Contract Design and Stability in Matching Markets” (Hatfield–K.) Scott Duke Kominers (Harvard) October 12, 2010 23

  57. � � Frontiers of Matching Theory Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity is necessary for stability! Scott Duke Kominers (Harvard) October 12, 2010 24

  58. � � Frontiers of Matching Theory Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity or transferable utility is necessary for stability! Scott Duke Kominers (Harvard) October 12, 2010 24

  59. � � Frontiers of Matching Theory Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity or transferable utility is necessary for stability! Scott Duke Kominers (Harvard) October 12, 2010 24

  60. � � Frontiers of Matching Theory Cyclic Contract Sets g P f 1 : { y , x 2 } ≻ { x 1 , x 2 } ≻ ∅ � ������� y f 1 P f 2 : { x 2 , x 1 } ≻ ∅ x 1 x 2 P g : { y } ≻ ∅ f 2 Theorem Acyclicity or transferable utility is necessary for stability! “Stability and CE in Trading Networks” (Hatfield–K.–Nichifor–Ostrovsky–Westkamp) Scott Duke Kominers (Harvard) October 12, 2010 24

  61. Frontiers of Matching Theory Conclusion Acyclicity and substitutability are necessary and sufficient for (classical) matching theory... ...and at the outer frontiers, surprising structure arises. Scott Duke Kominers (Harvard) October 12, 2010 25

  62. 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? Scott Duke Kominers (Harvard) October 12, 2010 25

  63. 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 Scott Duke Kominers (Harvard) October 12, 2010 25

  64. Frontiers of Matching Theory Extra Slides Extra Slides Scott Duke Kominers (Harvard) October 12, 2010 26

  65. Frontiers of Matching Theory Extra Slides Related Literature Scott Duke Kominers (Harvard) October 12, 2010 27

  66. Frontiers of Matching Theory Extra Slides Related Literature Gale–Shapley (1962) Scott Duke Kominers (Harvard) October 12, 2010 27

  67. Frontiers of Matching Theory Extra Slides Related Literature Roth (1986) Gale–Shapley (1962) Scott Duke Kominers (Harvard) October 12, 2010 27

  68. Frontiers of Matching Theory Extra Slides Related Literature Hatfield–Milgrom (2005) Echenique–Oviedo (2006) Roth (1986) Gale–Shapley (1962) Scott Duke Kominers (Harvard) October 12, 2010 27

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