Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Deferred acceptance algorithms: history, theory, practice, and open questions

Abstract

The deferred acceptance algorithm proposed by Gale and Shapley (1962) has had a profound influence on market design, both directly, by being adapted into practical matching mechanisms, and, indirectly, by raising new theoretical questions. Deferred acceptance algorithms are at the basis of a number of labor market clearinghouses around the world, and have recently been implemented in school choice systems in Boston and New York City. In addition, the study of markets that have failed in ways that can be fixed with centralized mechanisms has led to a deeper understanding of some of the tasks a marketplace needs to accomplish to perform well. In particular, marketplaces work well when they provide thickness to the market, help it deal with the congestion that thickness can bring, and make it safe for participants to act effectively on their preferences. Centralized clearinghouses organized around the deferred acceptance algorithm can have these properties, and this has sometimes allowed failed markets to be reorganized.

This is a preview of subscription content, log in to check access.

References

  1. Abdulkadiroğlu A, Pathak PA and Roth AE (2005a). The New York city high school match. Am Econ Rev Papers Proc 95(2): 364–367

  2. Abdulkadiroğlu A, Pathak PA, Roth AE and Sönmez T (2005b). The Boston public school match. Am Econ Rev Papers Proc 95(2): 368–371

  3. Abdulkadiroğlu A, Pathak PA, Roth AE (2006a) Strategy-proofness versus efficiency in matching with indifferences: redesigning the NYC High School Match. Working paper, November 2006

  4. Abdulkadiroğlu A, Pathak PA, Roth AE, Sönmez T (2006b) Changing the Boston school choice mechanism. Working paper, January 2006

  5. Abdulkadiroğlu A and Sönmez T (1998). Random serial dictatorship and the core from random endowments in house allocation problems (joint with Atila). Econometrica 66: 689–701

  6. Abdulkadiroğlu A and Sönmez T (1999). House allocation with existing tenants. J Econ Theory 88: 233–260

  7. Abdulkadiroğlu A and Sönmez T (2002). School choice: a mechanism design approach. Am Econ Rev 93: 729–747

  8. Abeledo HG and Isaak G (1991). A characterization of graphs which assure the existence of stable matchings. Math Soc Sci 22: 93–96

  9. Abeledo HG and Rothblum UG (1995). Courtship and linear programming. Linear Algebra Appl 216: 111–124

  10. Abraham D, Blum A, Sandholm T (2007) Clearing algorithms for Barter exchange markets: enabling nationwide kidney exchanges. In: Proceedings of the ACM conference on electronic commerce (EC)

  11. Adachi H (2000). On a characterization of stable matchings. Econ Lett 68: 43–49

  12. Aldershof B and Carducci OM (1996). Stable matchings with couples. Discr Appl Math 68(1–2): 203–207

  13. Artemov G (2007) Matching and price competition: would personalized prices help? Int J Game Theory (forthcoming)

  14. Avery C, Jolls C, Posner RA and Roth AE (2001). The market for federal judicial law clerks. Univ Chic Law Rev 68(3): 793–902

  15. Avery C, Fairbanks A and Zeckhauser R (2003). The early admissions game: joining the elite. Harvard University Press, Cambridge

  16. Avery C, Jolls C, Posner RA, Roth AE (2007) The new market for federal judicial law clerks. Univ Chic Law Rev (forthcoming)

  17. Balinski M and Sönmez T (1999). A tale of two mechanisms: student placement. J Econ Theory 84: 73–94

  18. Becker GS (1981). A treatise on the family. Harvard University Press, Cambridge

  19. Blair C (1988). The lattice structure of the set of stable matchings with multiple partners. Math Oper Res 13: 619–628

  20. Bulow J and Levin J (2006). Matching and price competition. Am Econ Rev 96(3): 652–658

  21. Cechlarova K, Fleiner T, Manlove D (2005) The kidney exchange game. In: Zadnik-Stirn L, Drobne S (eds) Proc. SOR ’05, pp 77–83

  22. Chung K-S (2000). On the existence of stable roommate matchings. Games Econ Behav 33(2): 206–230

  23. Crawford VP (2008) The flexible-salary match: a proposal to increase the salary flexibility of the national resident matching program. J Econ Behav Organ (forthcoming)

  24. Crawford VP and Knoer EM (1981). Job matching with heterogeneous firms and workers. Econometrica 49: 437–450

  25. Damiano E, Li H and Suen W (2005). Unravelling of dynamic sorting. Rev Econ Stud 72(4): 1057–1076

  26. Demange G and Gale D (1985). The strategy structure of two-sided matching markets. Econometrica 53: 873–888

  27. Demange G, Gale D and Sotomayor M (1986). Multi-item auctions. J Polit Econ 94: 863–872

  28. Demange G, Gale D and Sotomayor M (1987). A further note on the stable matching problem. Discrete Appl Math 16: 217–222

  29. Dubins LE and Freedman DA (1981). Machiavelli and the gale–shapley algorithm. Am Math Monthly 88: 485–494

  30. Echenique F and Oviedo J (2004). Core many-to-one matchings by fixed point methods. J Econ Theory 115(2): 358–376

  31. Echenique F and Oviedo J (2006). A theory of stability in many-to-many matching markets. Theor Econ 1(2): 233–273

  32. Echenique F and Yenmez MB (2007). A solution to matching with preferences over colleagues. Games Econ Behav 59(1): 46–71

  33. Ehlers L and Klaus B (2006). Efficient priority rules. Games Econ Behav 55: 372–384

  34. Erdil A, Ergin H (2006) Two-sided matching with indifferences. Working paper

  35. Erdil A, Ergin H (2007) What’s the matter with tie-breaking? Improving efficiency in school choice. Am Econ Rev (forthcoming)

  36. Ergin H (2002). Efficient resource allocation on the basis of priorities. Econometrica 70(6): 2489–2497

  37. Ergin H and Sonmez T (2006). Games of school choice under the Boston mechanism. J Public Econ 90: 215–237

  38. Fleiner T (2000) Stable and crossing structures. PhD Dissertation, Centrum voor Wiskunde en Informatica (CWI), Amsterdam. http://www.renyi.hu/~fleiner/dissertation.pdf

  39. Fleiner T (2003). A fixed-point approach to stable matchings and some applications. Math Oper Res 28(1): 103–126

  40. Fréchette G, Roth AE, Utku Ünver M (2007) Unraveling yields inefficient matchings: evidence from post-season college football bowls. Rand J Econ (forthcoming)

  41. Gale D and Shapley L (1962). College admissions and the stability of marriage. Am Math Monthly 69: 9–15

  42. Guillen P, Kesten O (2007) On-campus housing: theory vs. experiment. Working paper, April, Carnegie Mellon University

  43. Gusfield D and Irving RW (1989). The stable marriage problem: structure and algorithms. MIT Press, Cambridge

  44. Hatfield JW, Kojima F (2007) Matching with contracts: corrigendum. June, working paper

  45. Hatfield JW and Milgrom P (2005). Matching with contracts. Am Econ Rev 95(4): 913–935

  46. Immorlica N and Mahdian M (2005). Marriage, honesty and stability. SODA 2005: 53–62

  47. Irving RW (1985). An efficient algorithm for the “stable roommates” problem. J Algorithms 6: 577–595

  48. Irving RW (1998) Matching medical students to pairs of hospitals: a new variation on an old theme. In: Proceedings of ESA’98, the sixth annual European symposium on algorithms, Venice, Italy, 1998. Lecture Notes in Computer Science, vol 1461. Springer, Berlin, pp 381–392

  49. Irving RW, Manlove DF, Scott S (2000) The hospitals/residents problem with ties Lecture Notes in Computer Science 1851:259–271

  50. Kagel JH and Roth AE (2000). The dynamics of reorganization in matching markets: a laboratory experiment motivated by a natural experiment. Quart J Econ February: 201–235

  51. Kamecke U (1998). Wage formation in a centralized matching market. Int Econ Rev 39(1): 33–53

  52. Kelso AS Jr and Crawford VP (1982). Job matching, coalition formation and gross substitutes. Econometrica 50: 1483–1504

  53. Kesten O (2004) Student placement to public schools in US: two new solutions. Working paper, Carnegie Mellon University

  54. Kesten O (2006). On two competing mechanisms for priority based allocation problems. J Econ Theory 127: 155–171

  55. Klaus B and Klijn F (2005). Stable matchings and preferences of couples. J Econ Theory 121(1): 75–106

  56. Knuth DE (1976) Mariages stables, Montreal, Les Presses de l’Universite de Montreal

  57. Kojima F (2007). Matching and price competition: comment. Am Econ Rev 97(3): 1027–1031

  58. Kojima F, Pathak P (2007) Incentives and stability in large two-sided matching markets. Working paper

  59. Li H and Rosen S (1998). Unraveling in matching markets. Am Econ Rev 88(3): 371–387

  60. Li H and Suen W (2000). Risk sharing, sorting and early contracting. J Polit Econ 108(5): 1058–1091

  61. Li H and Suen W (2004). Self-fulfilling early-contracting rush. Int Econ Rev 45(1): 301–324(24)

  62. Manlove DF, Irving R, Iwama K, Miyazaki S and Morita Y (2002). Hard variants of stable marriage. Theor Comput Sci 276(1–2): 261–279

  63. Martínez R, Massó J, Neme A and Oviedo J (2001). On the lattice structure of the set of stable matchings for a many-to-one model. Optimization 50: 439–457

  64. Martínez R, Massó J, Neme A and Oviedo J (2004). An algorithm to compute the full set of many-to-many stable matching. Math Soc Sci 47: 187–210

  65. McKinney CN, Niederle M and Roth AE (2005). The collapse of a medical labor clearinghouse (and why such failures are rare). Am Econ Rev 95(3): 878–889

  66. McVitie DG and Wilson LB (1970). Stable marriage assignments for unequal sets. BIT 10: 295–309

  67. McVitie DG and Wilson LB (1971). The stable marriage problem. Commun ACM 14(7): 486–493

  68. Milgrom P (2004). Putting auction theory to work. Cambridge University Press, Cambridge

  69. Mullin FJ and Stalnaker JM (1952). The matching plan for internship placement: a report of the first year’s experience. J Med Educ 27: 193–200

  70. Niederle M (2007) Competitive wages in a match with ordered contracts. Am Econ Rev (forthcoming)

  71. Niederle M and Roth AE (2003a). Relationship between wages and presence of a match in medical fellowships. JAMA J Am Med Assoc 290(9): 1153–1154

  72. Niederle M and Roth AE (2003b). Unraveling reduces mobility in a labor market: gastroenterology with and without a centralized match. J Polit Econ 111(6): 1342–1352

  73. Niederle M and Roth AE (2004). The gastroenterology fellowship match: how it failed and why it could succeed once again. Gastroenterology 127(2): 658–666

  74. Niederle M and Roth AE (2005). The gastroenterology fellowship market: should there be a match?. Am Econ Rev Papers Proc 95(2): 372–375

  75. Niederle M, Roth AE (2008) The effects of a central clearinghouse on job placement, wages, and hiring practices: gastroenterology fellows as a case study. NBER (forthcoming)

  76. Niederle M, Yariv L (2007) Matching through Decentralized Markets. Working paper, Stanford University

  77. Niederle M, Proctor DD and Roth AE (2006). What will be needed for the new GI fellowship match to succeed. Gastroenterology 130: 218–224

  78. Ostrovsky M (2007) Stability in supply chain networks, working paper (revised), Stanford University GSB

  79. Papai S (2000). Strategyproof assignment by hierarchical exchange. Econometrica 68: 1403–1433

  80. Pathak P (2007) Lotteries in student assignment. Working paper

  81. Pathak P, Sönmez T (2007) Leveling the playing field: sincere and strategic players in the boston mechanism. Working paper

  82. Public Law 108–218; Pension funding equity Act of 2004, SEC. 207. http://frwebgate.access.gpo.gov/cgi-bin/getdoc.cgi?dbname=108_cong_public_laws&docid=f:publ218.108.pdf

  83. Pycia M (2007) Many-to-one matching with complementarities and peer effects. April, http://www.econ.psu.edu/~pycia/pycia-matching-with-complementarities-April30–07.pdf

  84. Ronn E (1990). NP-complete stable matching problems. J Algorithms 11(2): 285–304

  85. Roth AE (1982a). The economics of matching: stability and incentives. Math Oper Res 7: 617–628

  86. Roth AE (1982b). Incentive compatibility in a market with indivisible goods. Econ Lett 9: 127–132

  87. Roth AE (1984). The evolution of the labor market for medical interns and residents: a case study in game theory. J Polit Econ 92: 991–1016

  88. Roth AE (1985). The college admissions problem is not equivalent to the marriage problem. J Econ Theory 36: 277–288

  89. Roth AE (1986). On the allocation of residents to rural hospitals: a general property of two–sided matching markets. Econometrica 54: 425–427

  90. Roth AE (1990). New physicians: a natural experiment in market organization. Science 250: 1524–1528

  91. Roth AE (1991). A natural experiment in the organization of entry level labor markets: regional markets for new physicians and surgeons in the U.K. Am Econ Rev 81: 415–440

  92. Roth AE (2002). The economist as engineer: game theory, experimental economics and computation as tools of design economics. Fisher Schultz lecture. Econometrica 70(4): 1341–1378

  93. Roth AE (2003). The origins, history and design of the resident match. JAMA J Am Med Assoc 289(7): 909–912

  94. Roth AE and Peranson E (1999). The redesign of the matching market for American physicians: some engineering aspects of economic design. Am Econ Rev 89(4): 748–780

  95. Roth AE and Sotomayor M (1988). Interior points in the core of two-sided matching markets. J Econ Theory 45: 85–101

  96. Roth AE and Sotomayor M (1989). The college admissions problem revisited. Econometrica 57: 559–570

  97. Roth AE and Sotomayor M (1990). Two-sided matching: a study in game-theoretic modeling and analysis, econometric society monograph series. Cambridge University Press, Cambridge

  98. Roth AE and Vande Vate JH (1990). Random paths to stability in two-sided matching. Econometrica 58: 1475–1480

  99. Roth AE and Xing X (1994). Jumping the gun: imperfections and institutions related to the timing of market transactions. Am Econ Rev 84: 992–1044

  100. Roth AE and Xing X (1997). Turnaround time and bottlenecks in market clearing: decentralized matching in the market for clinical psychologists. J Polit Econ 105: 284–329

  101. Roth AE, Sönmez T and Utku Ünver M (2004). Kidney exchange. Quart J Econ 119(2): 457–488

  102. Roth AE, Sönmez T and Utku Ünver M (2005). A kidney exchange clearinghouse in New England. Am Econ Rev Papers Proc 95(2): 376–380

  103. Roth AE, Sönmez T and Utku Ünver M (2007). Efficient kidney exchange: coincidence of wants in a market with compatibility-based preferences. Am Econ Rev 97(3): 828–851

  104. Shapley LS and Shubik M (1972). The assignment game I: the core. Int J Game Theory 1: 111–130

  105. Shapley LS and Scarf H (1974). On cores and indivisibility. J Math Econ 1: 23–28

  106. Sönmez T (1997). Manipulation via capacities in two-sided matching markets. J Econ Theory 77(1): 197–204

  107. Sönmez T (1999). Can pre-arranged matches be avoided in two-sided matching markets. J Econ Theory 86: 148–156

  108. Sönmez T and Utku Ünver M (2005). House allocation with existing tenants: an equivalence. Games Econ Behav 52: 153–185

  109. Sotomayor M (1996). A non constructive elementary proof of the existence of stable marriages. Games Econ Behav 13: 135–7

  110. Sotomayor M (2000). Existence of stable outcomes and the lattice property for a unified matching market. Math Soc Sci 39: 119–132

  111. Sotomayor M (2007). Connecting the cooperative and competitive structures of the multiple-partners assignment game. J Econ Theory 134(1): 155–174

  112. Tan JJM (1991). A necessary and sufficient condition for the existence of a complete stable matching. J Algorithms 12(1): 154–178

  113. Tarski A (1955). A lattice theoretical fixpoint theorem and its applications. Pacific J Math 5: 2

  114. Ünver MU (2001). Backward unraveling over time: the evolution of strategic behavior in the entry-level British medical labor markets. J Econ Dyn Control 25: 1039–1080

Download references

Author information

Correspondence to Alvin E. Roth.

Additional information

Prepared for Gale’s Feast: a Day in honor of the 85th birthday of David Gale, July 2007, Stony Brook.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Roth, A.E. Deferred acceptance algorithms: history, theory, practice, and open questions. Int J Game Theory 36, 537–569 (2008). https://doi.org/10.1007/s00182-008-0117-6

Download citation

Keywords

  • Matching
  • Market design
  • Gale-shapley
  • Deferred acceptance