Skip to main content

Properties of solutions of cooperative games with transferable utilities

Abstract

We understand a solution of a cooperative TU-game as the α-prenucleoli set, αR, which is a generalization of the notion of the [0, 1]-prenucleolus. We show that the set of all α-nucleoli takes into account the constructive power with the weight α and the blocking power with the weight (1 − α) for all possible values of the parameter α. The further generalization of the solution by introducing two independent parameters makes no sense. We prove that the set of all α-prenucleoli satisfies properties of duality and independence with respect to the excess arrangement. For the considered solution we extend the covariance propertywith respect to strategically equivalent transformations.

This is a preview of subscription content, access via your institution.

References

  1. Scarf, H. “The Core of an N person game”, Econometrica, No. 35, 50–69 (1967).

    MathSciNet  Article  MATH  Google Scholar 

  2. Schmeidler, D. “The Nucleolus of a Characteristic Function Game”, SIAM J. Appl. Math., No. 17, 1163–1170 (1969).

    MathSciNet  Article  MATH  Google Scholar 

  3. Shapley, L. S. “A Value for n-Person Pames”, Contrib. Theory of Games, II, Ann. Math. Stud., No. 28, 307–317 (1953).

    MathSciNet  Google Scholar 

  4. Sudhölter, P. “TheModified Nucleolus: Properties and Axiomatizations”, Int. J. Game Theory, No. 26, 147–182 (1997).

    Article  MATH  Google Scholar 

  5. Tarashnina, S. “The Simplified Modified Nucleolus of a Cooperative TU-game”, TOP 19 (1), 150–166 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  6. Smirnova, N. V. and Tarashnina, S. I. “On a Generalization of the N-Nucleus in Cooperative Games”, Diskretn. Anal. Issled. Oper. 18, No. 4, 77–93 (2011).

    MathSciNet  MATH  Google Scholar 

  7. Smirnova, N. V. and Tarashnina, S. I. “Geometric Properties of the [0, 1]-Nucleolus in Cooperative TUGames”, Mat. Teor. Igr Prilozh. 4, No. 1, 55–73 (2012) [in Russian].

    MATH  Google Scholar 

  8. Maschler, M. “The Bargaining Set, Kernel, and Nucleolus: A Survey”, in R. J. Aumann, S. Hart (Eds.) Handbook of Game Theory with Economic Applications, Vol. 1. (Elsevier, Amsterdam, 1992), pp. 591–665.

    Chapter  Google Scholar 

  9. Pecherskii, S. L., Yanovskaya, E. E. Cooperative Games: Solutions and Axioms (Evrop. Univ. Press, Saint-Petersburg, 2004) [in Russian].

    Google Scholar 

  10. Peleg, B., Sudhölter, P. Introduction to the Theory of Cooperative Games (Kluwer Acad. Publ., Boston, Dordrecht, London, 2003).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. V. Smirnova.

Additional information

Original Russian Text © N.V. Smirnova, S.I. Tarashnina, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 6, pp. 73–85.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Smirnova, N.V., Tarashnina, S.I. Properties of solutions of cooperative games with transferable utilities. Russ Math. 60, 63–74 (2016). https://doi.org/10.3103/S1066369X16060086

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X16060086

Keywords

  • TU-game
  • prenucleolus
  • SM-nucleolus
  • [0, 1]-prenucleolus
  • α-prenucleoli set
  • duality