Introduction to the theory of choice and stable contracts

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Abstract

The paper is devoted to the presentation of the basic concepts and results of the theory of stable contract systems. This theory originated in 1962 and has significantly been developed since then. The main results (existence, polarization, latticing) were obtained in a bipartite situation, when contracting agents are divided into two groups, and contracts are concluded between agents from opposite groups. Another important limitation is that the agents' preferences are described by so-called Plott choice functions. The first part of the paper is devoted to this concept, which generalizes the concept of partial order. The second part sets out the theory of stable contracts itself.

About the authors

Vladimir Ivanovich Danilov

Central Economics and Mathematics Institute of the Russian Academy of Sciences

Author for correspondence.
Email: vdanilov43@mail.ru
Doctor of physico-mathematical sciences, Main Scientist Researcher

References

  1. R. Aharoni, T. Fleiner, “On a lemma of Scarf”, J. Combin. Theory. Ser. B, 87:1 (2003), 72–80
  2. М. А. Айзерман, А. В. Малишевский, “Некоторые аспекты общей теории выбора лучших вариантов”, Автомат. и телемех., 1981, № 2, 65–83
  3. A. Alkan, “A class of multipartner matching markets with a strong lattice structure”, Econom. Theory, 19:4 (2002), 737–746
  4. A. Alkan, D. Gale, “Stable schedule matching under revealed preference”, J. Econom. Theory, 112:2 (2003), 289–306
  5. L. Beklemishev, D. Gabelaia, “Topological interpretations of provability logic”, Leo Esakia on duality in modal and intuitionistic logics, Outst. Contrib. Log., 4, Springer, Dordrecht, 2014, 257–290
  6. C. Blair, “Every finite distributive lattice is a set of stable matchings”, J. Combin. Theory Ser. A, 37:3 (1984), 353–356
  7. C. Blair, “The lattice structure of the set of stable matchings with multiple partners”, Math. Oper. Res., 13:4 (1988), 619–628
  8. V. Danilov, G. Koshevoy, “Mathematics of Plott choice functions”, Math. Social Sci., 49:3 (2005), 245–272
  9. V. Danilov, G. Koshevoy, E. Savaglio, “Hyper-relations, choice functions, and orderings of opportunity sets”, Soc. Choice Welf., 45:1 (2015), 51–69
  10. M. Dickmann, N. Schwartz, M. Tressl, Spectral spaces, New Math. Monogr., 35, Cambridge Univ. Press, Cambridge, 2019, xvii+633 pp.
  11. J.-P. Doignon, J.-Cl. Falmagne, Knowledge spaces, Springer-Verlag, Berlin, 1999, xvi+333 pp.
  12. P. H. Edelman, “Meet-distributive lattices and the anti-exchange closure”, Algebra Universalis, 10:3 (1980), 290–299
  13. P. H. Edelman, R. E. Jamison, “The theory of convex geometries”, Geom. Dedicata, 19:3 (1985), 247–270
  14. J. Faenza, Xuan Zhang, “Affinely representable lattices, stable matchings, and choice functions”, Math. Program., 197:2, Ser. B (2023), 721–760
  15. T. Fleiner, “A fixed-point approach to stable matchings and some applications”, Math. Oper. Res., 28:1 (2003), 103–126
  16. D. Gale, L. S. Shapley, “College admissions and the stability of marriage”, Amer. Math. Monthly, 69:1 (1962), 9–15
  17. D. Gusfield, R. W. Irving, The stable marriage problem: structure and algorithms, Found. Comput. Ser., MIT Press, Cambridge, MA, 1989, xviii+240 pp.
  18. G. Z. Gutin, P. R. Neary, A. Yeo, “Unique stable matchings”, Games Econom. Behav., 141 (2023), 529–547
  19. Yuang-Cheh Hsueh, “A unifying approach to the structure of the stable matching problems”, Comput. Math. Appl., 22:6 (1991), 13–27
  20. R. W. Irving, “An efficient algorithm for the ‘stable roommates’ problem”, J. Algorithms, 6:4 (1985), 577–595
  21. A. V. Karzanov, Stable matchings, choice functions, and linear orders, 2025 (v1 – 2024), 26 pp.
  22. A. S. Kelso, Jr., V. P. Crawford, “Job matching, coalition formation, and gross substitutes”, Econometrica, 50:6 (1982), 1483–1504
  23. F. Klijn, A. Yazici, “A many-to-many ‘rural hospital theorem’ ”, J. Math. Econom., 54 (2014), 63–73
  24. D. E. Knuth, Mariages stables, Collection “Chaire Aisenstadt”, Les Presses de l'Universite de Montreal, Montreal, QC, 1976, 106 pp.
  25. B. Korte, L. Lovasz, R. Schrader, Greedoids, Algorithms Combin., 4, Springer-Verlag, Berlin, 1991, viii+211 pp.
  26. G. A. Koshevoy, “Choice functions and abstract convex geometries”, Math. Social Sci., 38:1 (1999), 35–44
  27. D. Lehmann, “Nonmonotonic logics and semantics”, J. Logic Comput., 11:2 (2001), 229–256
  28. T. Leinster, Higher operads, higher categories, London Math. Soc. Lecture Note Ser., 298, Cambridge Univ. Press, Cambridge, 2004, xiv+433 pp.
  29. Л. Ловас, М. Пламмер, Прикладные задачи теории графов. Теория паросочетаний в математике, физике, химии, Мир, М., 1998, 653 с.
  30. D. G. McVitie, L. B. Wilson, “The stable marriage problem”, Comm. ACM, 14:7 (1971), 486–490
  31. C. R. Plott, “Path independence, rationality, and social choice”, Econometrica, 41:6 (1973), 1075–1091
  32. A. E. Roth, “The evolution of the labor market for medical interns and residents: a case study in game theory”, J. Polit. Econ., 92:6 (1984), 991–1016
  33. A. E. Roth, M. A. Oliveira Sotomayor, Two-sided matching. A study in game-theoretic modeling and analysis, Econom. Soc. Monogr., 18, Cambridge Univ. Press, Cambridge, 1990, xiv+265 pp.
  34. J. J. M. Tan, “A necessary and sufficient condition for the existence of a complete stable matching”, J. Algorithms, 12:1 (1991), 154–178

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