Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Cooperative Games (von Neumann–Morgenstern Stable Sets)

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_99-2

Definition of the Subject

The von Neumann–Morgenstern stable set (hereafter stable set) is the first solution concept in cooperative game theory defined by J. von Neumann and O. Morgenstern. Though it was defined cooperative games in characteristic function form, von Neumann and Morgenstern gave a more general definition of a stable set in abstract games. Later, J. Greenberg and M. Chwe cleared a way to apply the stable set concept to the analysis of noncooperative games in strategic and extensive forms. Stable sets in a characteristic function form game may not exist, as was shown by W. F. Lucas for a ten-person game that does not admit a stable set. On the other hand, stable sets exist in many important games. In voting games, for example, stable sets exist, and they indicate what coalitions can be formed in detail. The core, on the other hand, can be empty in voting games, though it is one of the best known solution concepts in cooperative game theory. The analysis of stable sets is...


Coalition Structure Preference Profile Assignment Game External Stability Transferable Utility Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access


  1. Aumann R, Peleg B (1960) Von Neumann-Morgenstern solutions to cooperative games without side payments. Bull Am Math Soc 66:173–179MATHMathSciNetCrossRefGoogle Scholar
  2. Bala V, Goyal S (2000) A noncooperative model of network formation. Econometric 638:1181–1229MathSciNetCrossRefGoogle Scholar
  3. Baneijee S, Konishi H, Sonmez T (2001) Core in a simple coalition formation game. Soc Choice Welf 18:135–153MathSciNetCrossRefGoogle Scholar
  4. Barberà S, Gerber A (2003) On coalition formation: durable coalition structures. Math Soc Sci 45:185–203MATHCrossRefGoogle Scholar
  5. Beal S, Durieu J, Solal P (2008) Farsighted coalitional stability in TU-games. Math Soc Sci 56:303–313MATHMathSciNetCrossRefGoogle Scholar
  6. Bemheim D, Peleg B, Whinston M (1987) Coalition-proof Nash equilibria. J Econ Theory 42:1–12CrossRefGoogle Scholar
  7. Bhattacharya A, Brosi V (2011) An existence result for farsighted stable sets of games in characteristic function form. Int J Game Theory 40:393–401MATHMathSciNetCrossRefGoogle Scholar
  8. Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38:201–230MATHMathSciNetCrossRefGoogle Scholar
  9. Bott R (1953) Symmetric solutions to majority games. In: Kuhn HW, Tucker AW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 319–323Google Scholar
  10. Chwe MS-Y (1994) Farsighted coalitional stability. J Econ Theory 63:299–325ADSMATHMathSciNetCrossRefGoogle Scholar
  11. Diamantoudi E (2005) Stable cartels revisited. Econ Theory 26:907–921MATHMathSciNetCrossRefGoogle Scholar
  12. Diamantoudi E, Xue L (2003) Farsighted stability in hedonic games. Soc Choice Welf 21:39–61MATHMathSciNetCrossRefGoogle Scholar
  13. Diamantoudi E, Xue L (2007) Coalitions, agreements and efficiency. J Econ Theory 136:105–125MATHMathSciNetCrossRefGoogle Scholar
  14. Ehlers L (2007) Von Neumann-Morgenstem stable sets in matching problems. J Econ Theory 134:537–547MATHMathSciNetCrossRefGoogle Scholar
  15. Einy E, Shitovitz B (1996) Convex games and stable sets. Games Econ Behav 16:192–201MATHMathSciNetCrossRefGoogle Scholar
  16. Einy E, Holzman R, Monderer D, Shitovitz B (1996) Core and stable sets of large games arising in economics. J Econ Theory 68:200–211MATHMathSciNetCrossRefGoogle Scholar
  17. Funaki Y, Yamato T (2014) Stable coalition structures under restricted coalitional changes. Int Game Theory Rev 16 doi: 10.1142/S0219198914500066Google Scholar
  18. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15MATHMathSciNetCrossRefGoogle Scholar
  19. Greenberg J (1990) The theory of social situations: an alternative game theoretic approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  20. Greenberg J, Monderer D, Shitovitz B (1996) Multistage situations. Econometrica 64:1415–1437MATHMathSciNetCrossRefGoogle Scholar
  21. Greenberg J, Luo X, Oladi R, Shitovitz B (2002) (Sophisticated) stable sets in exchange economies. Games Econ Behav 39:54–70MATHMathSciNetCrossRefGoogle Scholar
  22. Griesmer JH (1959) Extreme games with three values. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 189–212Google Scholar
  23. Gusfield D, Irving R (1989) The stable marriage problem: structure and algorithms. MIT Press, BostonMATHGoogle Scholar
  24. Harsanyi J (1974) An equilibrium-point interpretation of stable sets and a proposed alternative definition. Manag Sci 20:1472–1495MATHMathSciNetCrossRefGoogle Scholar
  25. Hart S (1973) Symmetric solutions of some production economies. Int J Game Theory 2:53–62MATHCrossRefGoogle Scholar
  26. Hart S (1974) Formation of cartels in large markets. J Econ Theory 7:453–466CrossRefGoogle Scholar
  27. Heijmans J (1991) Discriminatory von Neumann-Morgenstem solutions. Games Econ Behav 3:438–452MATHMathSciNetCrossRefGoogle Scholar
  28. Herings JJ, Mauleon A, Vannetelbosch V (2009) Farsightedly stable networks. Games Econ Behav 67:526–541MATHMathSciNetCrossRefGoogle Scholar
  29. Herings JJ, Mauleon A, Vannetelbosch V (2010) Coalition formation among farsighted agents. Games 1:286–298MathSciNetCrossRefGoogle Scholar
  30. Jackson MO, van den Nouweland A (2005) Strongly stable networks. Games Econ Behav 51:420–444MATHCrossRefGoogle Scholar
  31. Jackson MO, Wolinsky A (1996) A strategic model of social and economic networks. J Econ Theory 71:44–74MATHMathSciNetCrossRefGoogle Scholar
  32. Kamijo Y, Muto S (2010) Farsighted coalitional stability of a price leadership cartel. Jpn Econ Rev 61:455–465MathSciNetCrossRefGoogle Scholar
  33. Kaneko M (1987) The conventionally stable sets in noncooperative games with limited observations I: definition and introductory argument. Math Soc Sci 13:93–128MATHCrossRefGoogle Scholar
  34. Kawasaki R (2010) Farsighted stability of the competitive allocations in an exchange economy with indivisible goods. Math Soc Sci 59:46–52MATHMathSciNetCrossRefGoogle Scholar
  35. Kawasaki R, Muto S (2009) Farsighted stability in provision of perfectly “lumpy” public goods. Math Soc Sci 58:98–109MATHMathSciNetCrossRefGoogle Scholar
  36. Klaus B, Klijn F, Walzl M (2010) Farsighted house allocation. J Math Econ 46:817–824MATHMathSciNetCrossRefGoogle Scholar
  37. Klaus B, Klijn F, Walzl M (2011) Farsighted stability for roommate markets. J Pub Econ Theory 13:921–933CrossRefGoogle Scholar
  38. Lucas WF (1968) A game with no solution. Bull Am Math Soc 74:237–239MATHCrossRefGoogle Scholar
  39. Lucas WF (1990) Developments in stable set theory. In: Ichiishi T et al (eds) Game theory and applications. Academic, New York, pp 300–316Google Scholar
  40. Lucas WF, Rabie M (1982) Games with no solutions and empty cores. Math Oper Res 7:491–500MATHMathSciNetCrossRefGoogle Scholar
  41. Lucas WF, Michaelis K, Muto S, Rabie M (1982) A new family of finite solutions. Int J Game Theory 11:117–127MATHMathSciNetCrossRefGoogle Scholar
  42. Luo X (2001) General systems and cp-stable sets: a formal analysis of socioeconomic environments. J Math Econ 36:95–109MATHCrossRefGoogle Scholar
  43. Luo X (2009) On the foundation of stability. Econ Theory 40:185–201ADSMATHCrossRefGoogle Scholar
  44. Manlove D (2013) Algorithmics of matching under preferences. World Scientific, SingaporeMATHCrossRefGoogle Scholar
  45. Mariotti M (1997) A model of agreements in strategic form games. J Econ Theory 74:196–217MATHMathSciNetCrossRefGoogle Scholar
  46. Masuda T (2002) Farsighted stability in average return games. Math Soc Sci 44:169–181MATHMathSciNetCrossRefGoogle Scholar
  47. Mauleon A, Vannetelbosch V, Vergote W (2011) VonNeumann -Morgenstem farsightedly stable sets in two-sided matching. Theoretical Econ 6:499–521MATHMathSciNetCrossRefGoogle Scholar
  48. Moulin H (1995) Cooperative microeconomics: a game-theoretic introduction. Princeton University Press, PrincetonCrossRefGoogle Scholar
  49. Muto S (1979) Symmetric solutions for symmetric constant-sum extreme games with four values. Int J Game Theory 8:115–123MATHMathSciNetCrossRefGoogle Scholar
  50. Muto S (1982a) On Hart production games. Math Oper Res 7:319–333MATHMathSciNetCrossRefGoogle Scholar
  51. Muto S (1982b) Symmetric solutions for (n, k) games. Int J Game Theory 11:195–201MATHMathSciNetCrossRefGoogle Scholar
  52. Muto S, Okada D (1996) Von Neumann-Morgenstem stable sets in a price-setting duopoly. Econ Econ 81:1–14Google Scholar
  53. Muto S, Okada D (1998) Von Neumann-Morgenstem stable sets in Cournot competition. Econ Econ 85:37–57Google Scholar
  54. Nakanishi N (1999) Reexamination of the international export quota game through the theory of social situations. Games Econ Behav 27:132–152MATHCrossRefGoogle Scholar
  55. Nakanishi N (2001) On the existence and efficiency of the von Neumann-Morgenstem stable set in an n-player prisoner’s dilemma. Int J Game Theory 30:291–307MATHCrossRefGoogle Scholar
  56. Nakanishi N (2009) Noncooperative farsighted stable set in an n-player prisoners’ dilemma. Int J Game Theory 38:249–261MATHCrossRefGoogle Scholar
  57. Núñez M, Rafels C (2013) Von Neumann-Morgenstem solutions in the assignment market. J Econ Theory 148:1282–1291MATHCrossRefGoogle Scholar
  58. Oladi R (2005) Stable tariffs and retaliation. Rev Int Econ 13:205–215CrossRefGoogle Scholar
  59. Owen G (1965) A class of discriminatory solutions to simple n-person games. Duke Math J 32:545–553MATHMathSciNetCrossRefGoogle Scholar
  60. Owen G (1968) n-Person games with only l, n-l, and n-person coalitions. Proc Am Math Soc 19:1258–1261MATHGoogle Scholar
  61. Owen G (1995) Game theory, 3rd edn. Academic, New YorkMATHGoogle Scholar
  62. Page FH, Wooders M (2009) Strategic basins of attraction, the path dominance core, and network formation games. Games Econ Behav 66:462–487MATHMathSciNetCrossRefGoogle Scholar
  63. Page FH, Wooders MH, Kamat S (2005) Networks and farsighted stability. J Econ Theory 120:257–269MATHMathSciNetCrossRefGoogle Scholar
  64. Peleg B (1986) A proof that the core of an ordinal convex game is a von Neumann-Morgenstem solution. Math Soc Sci 11:83–87MATHMathSciNetCrossRefGoogle Scholar
  65. Quint T, Wako J (2004) On house swapping, the strict core, segmentation, and linear programming. Math Oper Res 29:861–877MATHMathSciNetCrossRefGoogle Scholar
  66. Ray D, Vohra R (1997) Equilibrium binding agreements. J Econ Theory 73:30–78MATHMathSciNetCrossRefGoogle Scholar
  67. Rosenmüller J (1977) Extreme games and their solutions. Lecture notes in economics and mathematical systems, vol 145. Springer, BerlinGoogle Scholar
  68. Rosenmüller J, Shitovitz B (2000) A characterization of vNM-stable sets for linear production games. Int J Game Theory 29(3):9–61Google Scholar
  69. Roth AE, Postlewaite A (1977) Weak versus strong domination in a market with indivisible goods. J Math Econ 4:131–137MATHMathSciNetCrossRefGoogle Scholar
  70. Roth AE, Sotomayor MO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  71. Roth AE, Vande Vate JH (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480MATHMathSciNetCrossRefGoogle Scholar
  72. Serrano R, Volij O (2008) Mistakes in cooperation: the stochastic stability of Edgeworth’s recontracting. Econ J 118:1719–1741CrossRefGoogle Scholar
  73. Shapley LS (1953) Quota solutions of n-person games. In: Kuhn HW, Tucker TW (eds) Contribution to the theory of games, vol II. Annals of Mathematics Studies, vol 28. Princeton University Press, Princeton, pp 343–359Google Scholar
  74. Shapley LS (1959) The solutions of a symmetric market game. In: Tucker AW, Luce RD (eds) Contribution to the theory of games, vol IV. Annals of Mathematics Studies, vol 40. Princeton University Press, Princeton, pp 145–162Google Scholar
  75. Shapley LS (1962) Simple games: an outline of the descriptive theory. Behav Sci 7:59–66MathSciNetCrossRefGoogle Scholar
  76. Shapley LS (1964) Solutions of compound simple games. In: Tucker AW et al (eds) Advances in game theory. Annals of Mathematics Studies, vol 52. Princeton University Press, Princeton, pp 267–305Google Scholar
  77. Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26MATHMathSciNetCrossRefGoogle Scholar
  78. Shapley LS, Scarf H (1974) On cores and indivisibilities. J Math Econ 1:23–37MATHMathSciNetCrossRefGoogle Scholar
  79. Shapley LS, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1:111–130MathSciNetCrossRefGoogle Scholar
  80. Shino J, Kawasaki R (2012) Farsighted stable sets in Hotelling’s location games. Math Soc Sci 63:23–30MATHMathSciNetCrossRefGoogle Scholar
  81. Shitovitz B, Weber S (1997) The graph of Lindahl correspondence as the unique von Neumann-Morgenstem abstract stable set. J Math Econ 27:375–387MATHMathSciNetCrossRefGoogle Scholar
  82. Shubik M (1985) A game-theoretic approach to political economy. MIT Press, BostonGoogle Scholar
  83. Simonnard M (1966) Linear programming. Prentice-Hall, New JerseyMATHGoogle Scholar
  84. Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30:177–185MATHMathSciNetCrossRefGoogle Scholar
  85. Sung SC, Dimitrov D (2007) On myopic stability concepts for hedonic games. Theory Dec 62:31–45MATHMathSciNetCrossRefGoogle Scholar
  86. Suzuki A, Muto S (2000) Farsighted stability in prisoner’s dilemma. J Oper Res Soc Jpn 43:249–265ADSMATHMathSciNetGoogle Scholar
  87. Suzuki A, Muto S (2005) Farsighted stability in n-person prisoner’s dilemma. Int J Game Theory 33:431–445MATHMathSciNetCrossRefGoogle Scholar
  88. Suzuki A, Muto S (2006) Farsighted behavior leads to efficiency in duopoly markets. In: Haurie A et al (eds) Advances in dynamic games. Birkhauser, Boston, pp 379–395CrossRefGoogle Scholar
  89. Toda M (1997) Implementation and characterizations of the competitive solution with indivisibility. MimeoGoogle Scholar
  90. von Neumann J, Morgenstem O (1953) Theory of games and economic behavior, 3rd edn. Princeton University Press, PrincetonMATHGoogle Scholar
  91. Wako J (1984) A note on the strong core of a market with indivisible goods. J Math Econ 13:189–194MATHMathSciNetCrossRefGoogle Scholar
  92. Wako J (1991) Some properties of weak domination in an exchange market with indivisible goods. Jpn Econ Rev 42:303–314Google Scholar
  93. Wako J (1999) Coalitional-proofness of the competitive allocations in an indivisible goods market. Fields Inst Commun 23:277–283MathSciNetGoogle Scholar
  94. Wako J (2010) A polynomial-time algorithm to find von Neumann-Morgenstem stable matchings in marriage games. Algorithmica 58:188–220MATHMathSciNetCrossRefGoogle Scholar
  95. Xue L (1997) Nonemptiness of the largest consistent set. J Econ Theory 73:453–459MATHCrossRefGoogle Scholar
  96. Xue L (1998) Coalitional stability under perfect foresight. Econ Theory 11:603–627MATHCrossRefGoogle Scholar

Books and Reviews

  1. Lucas WF (1992) Von Neumann-Morgenstem stable sets. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. North-Holland, pp 543–590Google Scholar
  2. Shubik M (1982) Game theory in the social sciences: concepts and solutions. MIT Press, BostonMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Graduate School of Decision Science and TechnologyTokyo Institute of TechnologyTokyoJapan
  2. 2.Department of EconomicsGakushuin UniversityTokyoJapan