Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Cooperative Games (von Neumann–Morgenstern Stable Sets)

  • Ryo Kawasaki
  • Jun Wako
  • Shigeo Muto
Living reference work entry

Later version available View entry history

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...

Keywords

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.
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Bibliography

  1. Aumann R, Peleg B (1960) Von Neumann-Morgenstern solutions to cooperative games without side payments. Bull Am Math Soc 66:173–179zbMATHMathSciNetCrossRefGoogle 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 45:185–203zbMATHCrossRefGoogle Scholar
  5. Beal S, Durieu J, Solal P (2008) Farsighted coalitional stability in TU-games. Math Soc Sci 56:303–313zbMATHMathSciNetCrossRefGoogle 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 40:393–401zbMATHMathSciNetCrossRefGoogle Scholar
  8. Bogomolnaia A, Jackson MO (2002) The stability of hedonic coalition structures. Games Econ Behav 38:201–230zbMATHMathSciNetCrossRefGoogle 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 63:299–325ADSzbMATHMathSciNetCrossRefGoogle Scholar
  11. Diamantoudi E (2005) Stable cartels revisited. Econ Theory 26:907–921zbMATHMathSciNetCrossRefGoogle Scholar
  12. Diamantoudi E, Xue L (2003) Farsighted stability in hedonic games. Soc Choice Welf 21:39–61zbMATHMathSciNetCrossRefGoogle Scholar
  13. Diamantoudi E, Xue L (2007) Coalitions, agreements and efficiency. J Econ Theory 136:105–125zbMATHMathSciNetCrossRefGoogle Scholar
  14. Ehlers L (2007) Von Neumann-Morgenstem stable sets in matching problems. J Econ Theory 134:537–547zbMATHMathSciNetCrossRefGoogle Scholar
  15. Einy E, Shitovitz B (1996) Convex games and stable sets. Games Econ Behav 16:192–201zbMATHMathSciNetCrossRefGoogle Scholar
  16. Einy E, Holzman R, Monderer D, Shitovitz B (1996) Core and stable sets of large games arising in economics. J Econ Theory 68:200–211zbMATHMathSciNetCrossRefGoogle 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 69:9–15zbMATHMathSciNetCrossRefGoogle 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 64:1415–1437zbMATHMathSciNetCrossRefGoogle Scholar
  21. Greenberg J, Luo X, Oladi R, Shitovitz B (2002) (Sophisticated) stable sets in exchange economies. Games Econ Behav 39:54–70zbMATHMathSciNetCrossRefGoogle 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, BostonzbMATHGoogle Scholar
  24. Harsanyi J (1974) An equilibrium-point interpretation of stable sets and a proposed alternative definition. Manag Sci 20:1472–1495zbMATHMathSciNetCrossRefGoogle Scholar
  25. Hart S (1973) Symmetric solutions of some production economies. Int J Game Theory 2:53–62zbMATHCrossRefGoogle 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 3:438–452zbMATHMathSciNetCrossRefGoogle Scholar
  28. Herings JJ, Mauleon A, Vannetelbosch V (2009) Farsightedly stable networks. Games Econ Behav 67:526–541zbMATHMathSciNetCrossRefGoogle 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 51:420–444zbMATHCrossRefGoogle Scholar
  31. Jackson MO, Wolinsky A (1996) A strategic model of social and economic networks. J Econ Theory 71:44–74zbMATHMathSciNetCrossRefGoogle 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 13:93–128zbMATHCrossRefGoogle Scholar
  34. Kawasaki R (2010) Farsighted stability of the competitive allocations in an exchange economy with indivisible goods. Math Soc Sci 59:46–52zbMATHMathSciNetCrossRefGoogle Scholar
  35. Kawasaki R, Muto S (2009) Farsighted stability in provision of perfectly “lumpy” public goods. Math Soc Sci 58:98–109zbMATHMathSciNetCrossRefGoogle Scholar
  36. Klaus B, Klijn F, Walzl M (2010) Farsighted house allocation. J Math Econ 46:817–824zbMATHMathSciNetCrossRefGoogle 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 74:237–239zbMATHCrossRefGoogle 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 7:491–500zbMATHMathSciNetCrossRefGoogle Scholar
  41. Lucas WF, Michaelis K, Muto S, Rabie M (1982) A new family of finite solutions. Int J Game Theory 11:117–127zbMATHMathSciNetCrossRefGoogle Scholar
  42. Luo X (2001) General systems and cp-stable sets: a formal analysis of socioeconomic environments. J Math Econ 36:95–109zbMATHCrossRefGoogle Scholar
  43. Luo X (2009) On the foundation of stability. Econ Theory 40:185–201ADSzbMATHCrossRefGoogle Scholar
  44. Manlove D (2013) Algorithmics of matching under preferences. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
  45. Mariotti M (1997) A model of agreements in strategic form games. J Econ Theory 74:196–217zbMATHMathSciNetCrossRefGoogle Scholar
  46. Masuda T (2002) Farsighted stability in average return games. Math Soc Sci 44:169–181zbMATHMathSciNetCrossRefGoogle Scholar
  47. Mauleon A, Vannetelbosch V, Vergote W (2011) VonNeumann -Morgenstem farsightedly stable sets in two-sided matching. Theoretical Econ 6:499–521zbMATHMathSciNetCrossRefGoogle 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 8:115–123zbMATHMathSciNetCrossRefGoogle Scholar
  50. Muto S (1982a) On Hart production games. Math Oper Res 7:319–333zbMATHMathSciNetCrossRefGoogle Scholar
  51. Muto S (1982b) Symmetric solutions for (n, k) games. Int J Game Theory 11:195–201zbMATHMathSciNetCrossRefGoogle 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 27:132–152zbMATHCrossRefGoogle 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 30:291–307zbMATHCrossRefGoogle Scholar
  56. Nakanishi N (2009) Noncooperative farsighted stable set in an n-player prisoners’ dilemma. Int J Game Theory 38:249–261zbMATHCrossRefGoogle Scholar
  57. Núñez M, Rafels C (2013) Von Neumann-Morgenstem solutions in the assignment market. J Econ Theory 148:1282–1291zbMATHCrossRefGoogle 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 32:545–553zbMATHMathSciNetCrossRefGoogle Scholar
  60. Owen G (1968) n-Person games with only l, n-l, and n-person coalitions. Proc Am Math Soc 19:1258–1261zbMATHGoogle Scholar
  61. Owen G (1995) Game theory, 3rd edn. Academic, New YorkzbMATHGoogle Scholar
  62. Page FH, Wooders M (2009) Strategic basins of attraction, the path dominance core, and network formation games. Games Econ Behav 66:462–487zbMATHMathSciNetCrossRefGoogle Scholar
  63. Page FH, Wooders MH, Kamat S (2005) Networks and farsighted stability. J Econ Theory 120:257–269zbMATHMathSciNetCrossRefGoogle Scholar
  64. Peleg B (1986) A proof that the core of an ordinal convex game is a von Neumann-Morgenstem solution. Math Soc Sci 11:83–87zbMATHMathSciNetCrossRefGoogle Scholar
  65. Quint T, Wako J (2004) On house swapping, the strict core, segmentation, and linear programming. Math Oper Res 29:861–877zbMATHMathSciNetCrossRefGoogle Scholar
  66. Ray D, Vohra R (1997) Equilibrium binding agreements. J Econ Theory 73:30–78zbMATHMathSciNetCrossRefGoogle 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 4:131–137zbMATHMathSciNetCrossRefGoogle Scholar
  70. Roth AE, Sotomayor MO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  71. Roth AE, Vande Vate JH (1990) Random paths to stability in two-sided matching. Econometrica 58:1475–1480zbMATHMathSciNetCrossRefGoogle 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 1:11–26zbMATHMathSciNetCrossRefGoogle Scholar
  78. Shapley LS, Scarf H (1974) On cores and indivisibilities. J Math Econ 1:23–37zbMATHMathSciNetCrossRefGoogle 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 63:23–30zbMATHMathSciNetCrossRefGoogle Scholar
  81. Shitovitz B, Weber S (1997) The graph of Lindahl correspondence as the unique von Neumann-Morgenstem abstract stable set. J Math Econ 27:375–387zbMATHMathSciNetCrossRefGoogle 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 JerseyzbMATHGoogle Scholar
  84. Solymosi T, Raghavan TES (2001) Assignment games with stable core. Int J Game Theory 30:177–185zbMATHMathSciNetCrossRefGoogle Scholar
  85. Sung SC, Dimitrov D (2007) On myopic stability concepts for hedonic games. Theory Dec 62:31–45zbMATHMathSciNetCrossRefGoogle Scholar
  86. Suzuki A, Muto S (2000) Farsighted stability in prisoner’s dilemma. J Oper Res Soc Jpn 43:249–265ADSzbMATHMathSciNetGoogle Scholar
  87. Suzuki A, Muto S (2005) Farsighted stability in n-person prisoner’s dilemma. Int J Game Theory 33:431–445zbMATHMathSciNetCrossRefGoogle 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, PrincetonzbMATHGoogle Scholar
  91. Wako J (1984) A note on the strong core of a market with indivisible goods. J Math Econ 13:189–194zbMATHMathSciNetCrossRefGoogle 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 58:188–220zbMATHMathSciNetCrossRefGoogle Scholar
  95. Xue L (1997) Nonemptiness of the largest consistent set. J Econ Theory 73:453–459zbMATHCrossRefGoogle Scholar
  96. Xue L (1998) Coalitional stability under perfect foresight. Econ Theory 11:603–627zbMATHCrossRefGoogle 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, BostonzbMATHGoogle 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