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Review of the Theory of Stable Matchings and Contract Systems

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Abstract

A review of works devoted to the theory of stable matchings or, more generally, of stable networks of contracts is given. A set (network) of contracts is called stable if no coalition has an available contract that gives all coalition members strictly more than the proposed set. In a special case, this concept was introduced in 1962 by Gale and Shapley and has since gone a long way in its development both theoretically (theorems, structures, and algorithms) and in the field of applications in economics, physics, biology, and mathematics.

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REFERENCES

  1. M. A. Aizerman and F. T. Aleskerov, Choice of Variants: Foundations of the Theory (Nauka, Moscow, 1990) [in Russian].

    MATH  Google Scholar 

  2. F. T. Aleskerov and S. G. Kisel’gof, “Nobel prize winners 2012: Lloid Shapley and Elvin Roth,” Ekon. Zh. VShE 16 (4), 433–343 (2012).

    Google Scholar 

  3. L. Lovás and M. Plammer, Matching theory (North Holland, Amsterdam, 1986).

    Google Scholar 

  4. A. A. Vasin and V. A. Gurvich, “Reconcilable sets of coalitions for games in normal form,” Numerical Optimization Methods (SEI, Irkutsk, 1978).

    MATH  Google Scholar 

  5. V. I. Danilov, “On Scarf’s theorem,” Ekonom Mat. Metody 35 (3), 137-139 (1999).

    Google Scholar 

  6. V. I. Danilov, “Stable sysems of flexible contracts,” Zh. Novoi Ekon. Assotsiatsii 3 (51), 32–49 (2021).

    Google Scholar 

  7. S. G. Kisel’gof, “Modelling admission campaign: Universities of various quality and high school graduates with a quadratic utility function,” Probl. Upravl. 5, 33–40 (2012).

    Google Scholar 

  8. A. Abdulkadiroglu and T. Sonmez, “School choice: A mechanism design approach,” Am. Econ. Rev. 93, 729–747 (2003).

    Article  Google Scholar 

  9. H. G. Abeledo, Y. Blum, and U. G. Rothblum, “Canonical monotone decompositions of fractional stable matchings,” Int. J. Game Theory 25, 161–176 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Abeledo and G. Isaak, “A Characterization of graphs that ensure the existence of stable matchings,” Math. Social Sci. 22, 93–96 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Abraham, P. Biro, and D. Manlove, “'Almost stable' matchings in the roommates problem,” Approximation and Online Algorithms, Vol. 3879.

  12. K. Agoston, P. Biro, and I. McBride, “Integer programming methods for special college admissions problems,” J. Combin. Optim. 32, 1371–1399 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Adachi, “On a characterization of stable matchings,” Econ. Lett. 68, 43–49 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Aharoni and T. Fleiner, “On a lemma of Scarf,” J. Combin. Theory Ser. B 87 (1), 72–80 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Aldershof, O. Carducci, and D. Lorenc, “Refined inequalities for stable marriage,” Constraints 4 (3), 281–292 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Alimudin and Y. Ishida, “Matching-updating mechanism: A solution for the stable marriage problem with dynamic preferences,” Entropy 24, 263 (2022).

    Article  MathSciNet  Google Scholar 

  17. A. Alkan, “Nonexistence of stable threesome matchings,” Math. Social Sci. 16 (2), 207–209 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Alkan and D. Gale, “The core of the matching game,” Game Econ. Behavior 2, 203–212 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Alkan and D. Gale, “Stable schedule matching under revealed preference,” J. Econ. Theory 112 (2), 289–306 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Atay and M. Nuñez, “Multi-sided assignment games on m-partite graphs,” UB Economics Working Papers, 2017/357.

  21. E. M. Azevedo and J. W. Hatfield, “Existence of equilibrium in large matching markets with complementarities,” Available at SSRN, No. (2018), 3268884.

  22. H. Aziz, P. Biró, S. Gaspers, R. de Haan, N. Mattei, and B. Rastegari, “Stable matching with uncertain linear preferences,” Algorithmica 82, 1410–1433 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Aziz and B. Klaus, “Random matching under priorities: Stability and no envy concepts,” Social Choice Welfare 53 (2), 213–259 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Baïou and M. Balinski, “Many-to-many matching: Stable polyandrous polygamy (or polygamous polyandry),” Discr. Appl. Math. 101 (1–3), 1–12 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Baiou and M. Balinski, “The stable allocation (or ordinal transportation) problem,” Math. Oper. Res. 27 (3), 485–503 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Balinski and D. Gale, “On the core of the assignment game,” Game Theory and Applications. Economic Theory, Econometrics, and Mathematical Economics (1990), pp. 373–374.

    Book  MATH  Google Scholar 

  27. M. Balinski and G. Ratier, “Graphs and marriages,” Am. Math. Monthly 105 (5), 430–445 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Bando and T. Hirai, “Stability and venture structures in multilateral matching,” J. Econ. Theory 196, 105292 (2021).

  29. V. Bansal, A. Agrawal, and V. S. Malhotra, “Stable marriages with multiple partners: Efficient search for an optimal solution,” Proc. ICALP, 2003. Lect. Notes Comput. Sci. 2719, 527-542 (2003).

  30. A. Bhatnagar, V. Gambhir, and M. Thakur, “A new perspective to stable marriage problem in profit maximization of matrimonial websites,” J. Inform. Process. Syst. 14 (4), 961–979 (2018).

    Google Scholar 

  31. A. T. Benjamin, C. Converse, and H. A. Krieger, “How do i marry thee? Let me count the ways,” Discr. Appl. Math. 59, 285–292 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Biró, “The stable matching problem and its generalizations: An algorithmic and game theoretical approach,” PhD Thesis, Budapest, 2007.

  33. P. Biró and T. Fleiner, “Fractional solutions for capacitated NTU-Games, with applications to stable matchings,” Discr. Optim. 22 Part A, 241–254 (2016).

  34. P. Biró and F. Klijn, “Matching with couples: A multidisciplinary survey,” Int. Game Theory Rev. 15, 1340008 (2013).

  35. P. Biró and E. McDermid, “Three-sided stable matchings with cyclic preferences,” Algorithmica 58, 5–18 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  36. C. Blair, “Every finite distributive lattice is a set of stable matchings,” J. Combin. Theory, Ser. A 37, 353–356 (1984).

    MathSciNet  MATH  Google Scholar 

  37. C. Blair, “The lattice structure of the set of stable matchings with multiple partners,” Math. Oper. Res. 13 (4), 619–628 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  38. E. Boros, V. Gurvich, S. Jastar, and D. Krasner, “Stable matchings in three-sided systems with cyclic preferences,” Discr. Math. 289, 1–10 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  39. E. Boros, V. Gurvich, and A. Vasin, “Stable families of coalitions and normal hypergraphs,” Math. Social Sci. 34, 107–123 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  40. G. Caldarelli and A. Capocci, “Beauty and distance in the stable marriage problem,” Phys. A 300 (1–2), 325–331 (2001).

    Article  MATH  Google Scholar 

  41. D. Cantala, “Matching markets: The particular case of couples,” Econ. Bull. 3, 1–11 (2004).

    MathSciNet  MATH  Google Scholar 

  42. C. P. Chambers, “Consistency in the probabilistic assignment model,” J. Math. Econ. 40, 953–962 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  43. C. P. Chambers and M. B. Yenmez, “Choice and matching,” Am. Econ. J.: Microeconomics 9 (3), 126–147 (2017).

    Google Scholar 

  44. K. Cechlarova and T. Fleiner, “On a generalization of the stable roommates problem,” ACM Trans. Algorithms 1 (1), 143–156 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  45. Y.-K. Che, J. Kim, and F. Kojima, “Stable matching in large economies,” Econometrica 87 (1), 65–110 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Chowdhury, “Matching theory for cognitive radio networks: An overview,” ICT Express 5 (1), 12–15 (2019).

    Article  Google Scholar 

  47. K. Chung, “On the existence of stable roommate matchings,” Games Econ. Behavior 33, 206–230 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  48. O. Celik and V. Knoblauch, “Marriage matching with correlated preferences,” Working papers Univ. of Connecticut, Department of Economics, Nos. 2007-16 (2007).

  49. F. Cooper, “Fair and large stable matchings in the stable marriage and student–project allocation problems,” Thesis PhD (2020).

  50. L. Cui and W. Jia, “Cyclic stable matching for three-sided networking services,” Comput. Networks 57 (1), 351–363 (2013).

    Article  Google Scholar 

  51. V. Danilov, G. Koshevoy, and C. Lang, “Gross substitution, discrete convexity, and submodularity,” Discr. Appl. Math. 131 (2), 283–298 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  52. V. I. Danilov, “Existence of stable matchings in some three-sided systems,” Math. Social Sci. 46 (2), 145–148 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  53. V. I. Danilov, “Choice functions on posets,” arXiv:2101.11965 [math.CO].

  54. V. I. Danilov and G. A. Koshevoy, “Stable sets of contracts in two-sided markets,” arXiv:2108.06786 [math.CO].

  55. V. I. Danilov and A. V. Karzanov, “Stable and metastable contract networks,” arXiv:22202.13089 [math.CO].

  56. B. C. Dean and S. Munshi, “Faster algorithms for stable allocation problems,” Algorithmica 58, 59–81 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Delorme, S. Garcia, J. Gondzio, J. Kalcsics, D. Manlove, and W. Pettersson, “Mathematical models for stable matching problems with ties and incomplete lists,” Eur. J. Oper. Res. 277, 426–441 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  58. G. Demange, D. Gale, and M. Sotomayor, “A furter note on the stable matching problem,” Discr. Appl. Math. 16, 217–222 (1987).

    Article  MATH  Google Scholar 

  59. E. Drgas-Burchardt and Z. Switalski, “A number of stable matchings in models of the Gale–Shapley type,” Discr. Appl. Math. 161, (18), 2932–2936 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  60. L. Dubins and D. Freedman, “Machiavelli and the Gale–Shapley algorithm,” Am. Math. Monthly 88 (7), 485–494 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  61. M. Dzierzawa and M.-J. Omero, “Statistics of stable marriages,” Phys. A 287 (1–2), 321–333 (2000).

    Article  MathSciNet  Google Scholar 

  62. F. Echenique and J. Oviedo, “A theory of stability in many-to-many matching markets,” Theor. Econ. 1, 233–273 (2006).

    Google Scholar 

  63. F. Echenique and B. Yenmez, “A solution to matching with preferences over colleagues,” Games Econ. Behavior 59, 46–71 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  64. J. Eeckhout, “On the uniqueness of stable marriage matchings,” Econ. Lett. 69, 1–8 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  65. A. Eguchi, S. Fujishige, and A. Tamura, “A generalized Gale–Shapley algorithm for a discrete-concave stable-marriage model,” Algorithms and Computation, Lect. Notes Comput. Sci. 2906, 495–504 (2003).

    Book  MATH  Google Scholar 

  66. P. Eirinakis, D. Magos, I. Mourtos, and P. Miliotis, “Polyhedral aspects of stable marriage,” Math. Oper. Res. 39 (3), 656–671 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  67. P. Eirinakis, D. Magos, and I. Mourtos, “The stable b-matching polytope revisited,” Discr. Appl. Math. 250, 186–201 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  68. K. Eriksson, J. Sjostrand, and P. Strimling, “Three-dimensional stable matching with cyclic preferences,” Math. Social Sci. 52, 77–87 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  69. K. Eriksson and J. Karlander, “Stable outcomes of the roommate game with transferable utility,” Int. J. Game Theory 29 (4), 555–569 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  70. P. Everaere, M. Morge, and G. Picard, “Minimal concession strategy for reaching fair, optimal and stable marriages,” Proc. of the 2013 Int. Conference on Autonomous Agents and Multi-Agent Systems, Citeseer, 2013, pp. 1319–1320.

  71. R. Farooq, T. Fleiner, and A. Tamura, “Matching with partially ordered contracts,” Japan J. Industr. Appl. Math. 29, 401–417 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  72. T. Feder, “A New fixed point approach for stable networks and stable marriages,” J. Comput. Syst. Sci. 45, 233–284 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  73. E. Fenoaltea, I. Baybusinov, J. Zhao, L. Zhou, and Y.-C. Zhang, “The stable marriage problem: An interdisciplinary review from the physicist’s perspective,” Phys. Rep. 917, 1–79 (2021).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  75. T. Fleiner, “On the stable -matching polytope,” Math. Social Sci. 46 (2), 149–158 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  76. T. Fleiner, “The stable roommate problem with choice functions,” Algorithmica 58, 82–101 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  77. T. Fleiner, “On stable matchings and flows,” Algorithms 7, 1–14 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  78. T. Fleiner, R. Jagadeesan, Z. Janko, and A. Teytelboym, “Trading networks with frictions,” Econometrica 87 (5), 1633–1661 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  79. S. Fujishige and A. Tamura, “A general two-sided matching market with discrete concave utility functions,” Discr. Appl. Math. 154 (6), 950–970 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  80. D. Gale, “The two-sided matching problem: Origin, development and current issues,” Int. Game Theory Rev. 3 (2–3), 237–252 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  81. D. Gale and L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Monthly 69 (1), 9–15 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  82. D. Gale and M. Ms. Sotomayor, “Machiavelli and the stable matching problem,” Am. Math. Monthly 92 (4), 261–268 (1985).

    Article  MATH  Google Scholar 

  83. D. Gale and M. Sotomayor, “Some remarks on the stable matching problem,” Discr. Appl. Math. 11 (3), 223–232 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  84. M. Gelain, M. Pini, F. Rossi, K. Venable, and T. Walsh, “Procedural fairness in stable marriage problems,” Proc. of 10th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS), 2011, Ed. by Tumer, Yolum, Sonenberg, and Stone (Taipei, 2011), pp. 1209–1210 .

  85. I. Giannakopoulos, P. Karras, D. Tsoumakos, K. Doka, and N. Koziris, “An equitable solution to the stable marriage problem,” 27th Int. Conference on Tools with Artificial Intelligence (ICTAI), IEEE, 2015, pp. 989–996.

  86. Y. Gonczarowski, N. Nisan, R. Ostrovsky, and W. Rosenbaum, “A stable marriage requires communication,” Games Econ. Behavior 118, 626–647 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  87. A. Goyal, V. Dubinkina, and S. Maslov, “Multiple stable states in microbial communities explained by the stable marriage problem,” ISME J. 12 (12), 2823–2834 (2018).

    Article  Google Scholar 

  88. F. Gul and E. Stacchetti, “Walras equilibrium with gross substitutes,” J. Econ. Theory 87, 95–124 (1999).

    Article  MATH  Google Scholar 

  89. D. Gusfield and R. W. Irving, The Stable Marriage Problem: Structure and Algorithms (MIT, Press. MA, Boston 1989).

    MATH  Google Scholar 

  90. D. Gusfield, R. Irving, P. Leather, and M. Saks, “Every finite distributive lattice is a set of stable matchings for a small stable marriage instance,” J. Combin. Theory, Ser. A 44, 304–309 (1987).

    MathSciNet  MATH  Google Scholar 

  91. M. M. Halldórsson, R. W. Irving, K. Iwama, D. F. Manlove, S. Miyazaki, Y. Morita, and S. Scott, “Approximability results for stable marriage problems with ties,” Theor. Comput. Sci. 306, 431–447 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  92. M. M. Halldorsson, K. Iwama, S. Miyazaki, and H. Yanagisawa, “Randomized Approximation of the Stable Marriage Problem,” Theor. Comput. Sci. 325 (3), 439–465 (2004).

    Article  MATH  Google Scholar 

  93. M. M. Halldorsson, K. Iwama, S. Miyazaki, and H. Yanagisawa, “Improved approximation results of the stable marriage problem,” ACM Trans. Algorithms 3 (3), Article 30 (2007).

  94. J. Hatfield and P. Milgrom, “Matching with contracts,” Am. Econ. Rev. 95, 913–935 (2005).

    Article  Google Scholar 

  95. J. Hatfield and F. Kojima, “Substitutes and stability for matching with contracts,” J. Econ. Theory 145, 1704–1723 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  96. J. W. Hatfield and S. D. Kominers, “Matching in networks with bilateral contracts,” Am. Econ. J.: Microeconomics 4, 176–208 (2012).

    Google Scholar 

  97. J. W. Hatfield and S. D. Kominers, “Multilateral matching,” J. Econ. Theory, 156 (C), 175–206.

  98. J. W. Hatfield and S. D. Kominers, “Contract design and stability in many-to-many matching,” Games Econ. Behavior 101, 78–97 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  99. W. Hatfield, D. Scott, A. Nichifor, M. Ostrovsky, and A. Westkamp, “Stability and competitive equilibrium in trading networks,” J. Political Econ. 121 (5), 966–1005 (2013).

    Article  Google Scholar 

  100. D. Henderson, “On marriage, kidneys and the economics nobel,” Wall Street J., Oct. 15, 2012.

  101. J. Hidakatsu, “Structure of the stable marriage and stable roommate problems and applications,” Thesis, Univ. of South Carolina (2016).

  102. C. -C. Huang, “Two’s company, three’s a crowd: Stable family and threesome roommates problems,” European Symposium on Algorithms (Springer, 2007), pp. 558–569.

  103. C. Huang, “Stable matching: an integer programming approach,” arXiv:2103.03418v1 [econ.TH].

  104. C. Huang, “Unidirectional substitutes and complements,” arXiv:2108.12572v1 [econ.TH].

  105. C. -C. Huang, “Classified stable matching,” Proc. of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 2010, pp. 1235–1253.

  106. R. Irving, P. Leather, and D. Gusfield, “An efficient algorithm for the ‘optimal’ stable marriage,” J. ACM 34 (3), 532–543 (1987).

    Article  MathSciNet  Google Scholar 

  107. R. Irving, D. Manlove, and S. Scott, “The hospitals/residents problem with ties,” Scandinavian Workshop on Algorithm Theory, 2000 (Springer. 2000), pp. 259–271.

  108. R. Irving, “An Efficient Algorithm for the "Stable Roommates” Problem," J. Algorithms 6 (4), 577–595 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  109. R. Irving, “Stable marriage and indifference,” Discr. Appl. Math. 48, 261–272 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  110. R. Irving, “The cycle roommates problem: A hard case of kidney exchange,” Inform. Process. Lett. 103 (1), 1–4 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  111. R. Irving and D. Manlove, “The stable roommates problem with ties,” J. Algorithms 43 (1), 85–105 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  112. R. Irving and P. Leather, “The complexity of counting stable marriages,” SIAM J. Comput. 15 (3), 655–667 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  113. R. Irving and S. Scott, “The stable fixtures problem - a many-to-many extension of stable roommates,” Discr. Appl. Math. 155, 2118–2129 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  114. K. Iwama, S. Miyazaki, and K. Okamoto, “Stable Roommates Problem with Triple Rooms,” Proc. 10th Korea–Japan Joint Workshop on Algorithms and Computation (WAAC 2007), pp. 105–112.

  115. Y. Ishida, “Antibody-Based Computing: An Application To the Stable Marriage Problem,” Artif. Life Robot 12 (1–2), 125–128 (2008).

    Article  Google Scholar 

  116. K. Iwama and S. Miyazaki, “A survey of the stable marriage problem and its variants,” Int. Conference on Informatics Education and Research for Knowledge-Circulating Society, ICKS, 2008, pp. 131–136.

  117. M. Jackson and A. Watts, “Equilibrium existence in bipartite social games: A generalization of stable matchings,” Econ. Bull. 3 (12), 1–8 (2008).

    Google Scholar 

  118. N. Kamiyama, “A new approach to the Pareto stable matching problem,” Math. Oper. Res. 39 (3), 851–862 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  119. V. Kanade, N. Leonardos, and F. Magniez, “Stable matching with evolving preferences,” Leibniz International Proceedings in Informatics (LIPICS) 60, 36:1–36:13.

  120. M. Kaneko and M. H. Wooders, “Cores of partitioning games,” Math. Social Sci. 3 (4), 313–327 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  121. A. Karlin, S. Gharan, and R. Weber, “A simply exponential upper bound on the maximum number of stable matchings,” Proc. of the 50th Annual ACM SIGACT Symposium on Theory of Computing, 2018, pp. 920–925.

  122. A. V. Karzanov, “On stable flows and preflows,” arXiv:2209.00614 [math.CO], Sep 2022.

  123. A. Kelso and V. Crawford, “Job matching, coalition formation, and gross substitutes,” Econometrica 50 (6), 1483–1504 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  124. T. Kiraly and J. Pap, “Kernels, stable matchings, and Scarf’s lemma,” Egervary Research Group, TR-2008-13 (2008).

  125. S. Kiselgof, “Matchings with interval order preferences: Efficiency vs strategy-proofness,” Proc. Comput. Sci. 31, 807–813 (2014).

    Article  Google Scholar 

  126. B. Klaus and M. Walzl, “Stable many-to-many matchings with contracts,” J. Math. Econ. 45, 422–434 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  127. F. Klijn and A. Yazici, “A many-to-many 'rural hospital theorem',” J. Math. Econ. 54, 63–73 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  128. V. Knoblauch, “Marriage matching: A conjecture of Donald Knuth,” Univ. of Connecticut, Working Paper, no. 2007–15.

  129. D. Knuth, Mariages Stables (Les Presses de L’Universite de Montreal, 1976); Stable Marriage and Its Relation to other Combinatorial Problems (Am. Math. Soc., Providence, RI, 1997).

    Google Scholar 

  130. F. Kojima, “Finding all stable matchings with couples,” J. Dynam. Games 2 (2), 321–330 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  131. V. Komornik, Z. Komornik, and C. Viauroux, “Stable schedule matchings by a fixed point method,” Acta Math. Hungarica 135, 67–79 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  132. E. Kujansuu, T. Lindberg, and E. Mokinen, “The stable roommates problem and chess tournament pairings,” Divulg. Mat. 7 (1), 19–28 (1999).

    MathSciNet  MATH  Google Scholar 

  133. A. Lage-Castellanos and R. Mulet, “The marriage problem: From the bar of appointments to the agency,” Phys. A 364, 389–402 (2006).

    Article  Google Scholar 

  134. C.-K. Lam and C. G. Plaxton, “On the Existence of Three-Dimensional Stable Matchings with Cyclic Preferences,” Theory Comput. Syst. 66 (2), 679–695 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  135. E. Yu. Lerner and R. E. Lerner, “Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferences,” arXiv:2101.08223 [math.CO].

  136. L. Lovasz, “Normal hypergraphs and the perfect graph conjecture,” Discr. Math. 2 (3), 253–267 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  137. V. S. Malhotra, “On the stability of multiple partner stable marriages with ties,” Proc. ESA, 2004. Lect. Notes Comput. Sci. 3221, 508–519 (2004).

  138. D. F. Manlove, “Stable marriage with ties and unacceptable partners,” Univ. of Glasgow, Computing Science Department Research Report no. TR-1999-29 (1999).

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

    Article  MathSciNet  MATH  Google Scholar 

  140. R. Martinez, J. Masso, A. Neme, and J. Oviedo, “An algorithm to compute the full set of many-to-many stable matchings,” Math. Social Sci. 47 (2), 187–210 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  141. D. McVitie and L. Wilson, “The stable marriage problem,” Commun. ACM 14 (7), 486–490 (1971).

    Article  MathSciNet  Google Scholar 

  142. P. Neme and J. Oviedo, “A note on the lattice structure for matching markets via linear programming,” J. Dynam. Games 8 (1), 61–67 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  143. C. Ng and D. Hirschberg, “Three-dimensional stable matching problems,” SIAM J. Discr. Math. 4 (2), 245–252 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  144. M. Nuñez and C. Rafels, “A survey on assignment markets,” J. Dynam. Games 2 (3–4), 227-256 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  145. M.-J. Omero, M. Dzierzawa, M. Marsili, and Y.-C. Zhang, “Scaling behavior in the stable marriage problem,” 1997. arXiv:cond-mat/9708181v1.

  146. M. Ostrovsky, “Stability in supply chain network,” Am. Econ. Rev. 98, 897–923 (2006).

    Article  Google Scholar 

  147. R. Ostrovsky and W. Rosenbaum, “It’s not easy being three: The approximability of three-dimensional stable matching problems,” 2014. Preprint arXiv:1412.1130.

  148. N. Panchal and S. Sharma, “An efficient algorithm for three dimensional cyclic stable matching,” Int. J. Eng. Res. & Technol. 3 (4), 2539–2544 (2014).

    Google Scholar 

  149. B. Pittel, “The "stable roommates” problem with random preferences," Ann. Probab. 21 (3), 1441–1477 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  150. B. Pittel, “The average number of stable matchings,” SIAM J. Discr. Math. 2 (4), 530–549 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  151. B. Pittel, “On likely solutions of the stable matching problem with unequal numbers of men and women,” Math. Oper. Res. 44 (1), 122–146 (2019).

    MathSciNet  MATH  Google Scholar 

  152. C. Plott, “Path independence, rationality, and social choice,” Econometrica 41, 1075–1091 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  153. P. Prosser, “Stable roommates and constraint programming,” 11th Int. Conference, CPAIOR, 2014, Cork, Ireland, pp. 15–28.

  154. M. Pycia, “Stability and Preference Alignment in Matching and Coalition Formation,” Econometrica 80 (1), 323–362 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  155. G. Ratier, “On the stable marriage polytope,” Discr. Math. 148, 141–159 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  156. E. Ronn, “NP-complete stable matching problems,” J. Algorithms, No. 2, 285–304 (1990).

  157. M. Rostek and N. Yoder, “Matching with complementary contracts,” Econometrica 88 (5), 1793–1827 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  158. A. Roth, “Stability and polarization of interests in job matching,” Econometrica 52, 47–57 (1984).

    Article  MATH  Google Scholar 

  159. A. Roth, “On the allocation of residents to rural hospitals: A general property of two-sided matching markets,” Econometrica 54, 425–427 (1986).

    Article  MathSciNet  Google Scholar 

  160. A. Roth and M. Sotomayor, Two-sided matching: A Study in Game-theoretic Modeling and Analysis (Cambridge Univ. Press, Cambridge, 1991).

    MATH  Google Scholar 

  161. A. Roth, T. Sonmez, and M. Unver, “Kidney exchange,” Quart. J. Econ. 119 (2), 457–488 (2004).

    Article  MATH  Google Scholar 

  162. A. Roth, T. Sonmez, and M. Unver, “Pairwise kidney exchange,” J. Econ. Theory 125 (2), 151–188 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  163. A. Roth and J. Vande Vate, “Random paths to stability in two-sided matching,” Econometrica 58, 475–1480 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  164. A. Roth, U. G. Rothblum, and J. Vande Vate, “Stable matchings, optimal assignments, and linear programming,” Math. Oper. Res. 18, 803–828 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  165. S. Scott, “A study of stable marriage problems with ties,” Ph.D. Thesis, Univ. of Glasgow (2005).

  166. L. Shapley and M. Shubik, “The assignment game. I. The core,” Int. J. Game Theory 1 (2), 111–130 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  167. G.-Y. Shi, Y. Kong, B. Chen, G. Yuan, and R. Wu, “Instability in stable marriage problem: Matching unequally numbered men and women,” Complexity Article ID 7409397 (2018).

  168. M. Sotomayor, “Three remarks on the many-to-many stable matching problem,” Math. Social Sci. 38 (1), 55–70 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  169. M. Sotomayor, “My encounters with David Gale,” Games Econ. Behavior 66, 643–646 (2009).

    Article  MATH  Google Scholar 

  170. H. Stuart, “The supplier-firm-buyer game and its m-sided generalization,” Math. Social Sci. 34 21–27 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  171. A. Subramanian, “A new approach to stable matching problems,” SIAM J. Comput. 23 (4), 671–700 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  172. M. Szestopalow, “Properties of stable matchings,” Thesis, Waterloo, Ontario, Canada (2010).

  173. J. Tan, “A Necessary and sufficient condition for the existence of a complete stable matching,” J. Algorithms, No. 1, 154–178 (1991).

  174. C.-P. Teo and J. Sethuraman, “The Geometry of fractional stabe matchings and its applications,” Math. Oper. Res. 23 (4), 874–891 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  175. C. -P. Teo, J. Sethuraman, and W.-P. Tan, “Gale–Shapley stable marriage problem revisited: Strategic issues and applications,” Manag. Sci. 47 (9), 1252–1267 (2001).

    Article  MATH  Google Scholar 

  176. E. Thurber, “Concerning the maximum number of stable matchings in the stable marriage problem,” Discrete Math. 248 (1–3), 195–219 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  177. H. Tong, H. Liang, and F. Bai, “The multi-dimensional stable marriage problem and its application in chemistry” (2015).

  178. J. Vande Vate, “Linear programming brings marital bliss,” Oper. Res. Lett. 8 (3), 147–153 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  179. L. Wilson, “An analysis of the stable marriage assignment algorithm,” BIT Numer. Math. 12 (4), 569–575 (1972).

    Article  MATH  Google Scholar 

  180. H. Xu and B. Li, “Seen as stable marriages,” Proc. IEEE INFOCOM, 2011, pp. 586–590.

  181. L. Zhou, “On a conjecture by Gale about one-sided matching problems,” J. Econ. Theory 52, 123–135 (1990).

    Article  MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

I am grateful to N.S. Kukushkin for improving the formulation of Proposition 2.2.

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This work was supported by the Russian Foundation Research, project no. 20-010-00569-A.

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  1. Central Economics and Mathematics Institute, Russian Academy of Sciences, 117418, Moscow, Russia

    V. I. Danilov

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Danilov, V.I. Review of the Theory of Stable Matchings and Contract Systems. Comput. Math. and Math. Phys. 63, 466–490 (2023). https://doi.org/10.1134/S0965542523030065

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