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Core stability and other applications of minimal balanced collections

Author

Listed:
  • Sudhölter, Peter

    (Department of Economics)

  • Grabisch, Michel

    (Paris School of Economics, Université Paris)

  • Laplace Mermoud, Dylan

    (Paris School of Economics, Université Paris)

Abstract

We describe algorithms and their implementations as computer programs derived from several theoretical results of the theory of cooperative transferable utility (TU) games. We show how to use Peleg’s well-known inductive method to explicitly compute all minimal balanced collections of coalitions. The described method is of independent interest and applied in the implementations of (a) the Bondareva-Shapley Theorem, which allows checking whether a TU game is balanced, i.e., has a non-empty core, and (b) a recent result of the second and third author that provides a sufficient and necessary condition for the stability of the core, which allows checking whether a balanced TU game has a core that is a von Neumann-Morgenstern stable set.

Suggested Citation

  • Sudhölter, Peter & Grabisch, Michel & Laplace Mermoud, Dylan, 2022. "Core stability and other applications of minimal balanced collections," Discussion Papers on Economics 4/2022, University of Southern Denmark, Department of Economics.
    • Handle: RePEc:hhs:sdueko:2022_004
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    References listed on IDEAS

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    1. repec:hal:pseose:halshs-00950109 is not listed on IDEAS
    2. Grabisch, Michel & Sudhölter, Peter, 2014. "On the restricted cores and the bounded core of games on distributive lattices," European Journal of Operational Research, Elsevier, vol. 235(3), pages 709-717.
    3. Lloyd S. Shapley, 1967. "On balanced sets and cores," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(4), pages 453-460.
    4. Pulido, Manuel A. & Sanchez-Soriano, Joaquin, 2006. "Characterization of the core in games with restricted cooperation," European Journal of Operational Research, Elsevier, vol. 175(2), pages 860-869, December.
    5. Bezalel Peleg, 1965. "An inductive method for constructing mimmal balanced collections of finite sets," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 12(2), pages 155-162, June.
    6. Lucas, William F., 1992. "Von Neumann-Morgenstern stable sets," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 17, pages 543-590, Elsevier.
    7. Biswas, A. K. & Parthasarathy, T. & Potters, J. A. M. & Voorneveld, M., 1999. "Large Cores and Exactness," Games and Economic Behavior, Elsevier, vol. 28(1), pages 1-12, July.
    8. Xiaotie Deng & Christos H. Papadimitriou, 1994. "On the Complexity of Cooperative Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 257-266, May.
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    Cited by:

    1. Dylan Laplace Mermoud, 2023. "Geometry of Set Functions in Game Theory: Combinatorial and Computational Aspects," Papers 2301.02950, arXiv.org, revised Oct 2023.

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    More about this item

    Keywords

    Core; stable set; minimal balanced collections; cooperative game.;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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