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Characterization of TU games with stable cores by nested balancedness

Author

Listed:
  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Peter Sudhölter

    (SDU - University of Southern Denmark)

Abstract
A balanced transferable utility game (N, v) has a stable core if its core is externally stable, that is, if each imputation that is not in the core is dominated by some core element. Given two payoff allocations x and y, we say that x outvotes y via some coalition S of a feasible set if x dominates y via S and x allocates at least v(T) to any feasible T that is not contained in S. It turns out that outvoting is transitive and the set M of maximal elements with respect to outvoting coincides with the core if and only if the game has a stable core. By applying the duality theorem of linear programming twice, it is shown that M coincides with the core if and only if a certain nested balancedness condition holds. Thus, it can be checked in finitely many steps whether a balanced game has a stable core. We say that the game has a super-stable core if each payoff vector that allocates less than v(S) to some coalition S is dominated by some core element and prove that core super-stability is equivalent to vital extendability, requiring that each vital coalition is extendable.

Suggested Citation

  • Michel Grabisch & Peter Sudhölter, 2024. "Characterization of TU games with stable cores by nested balancedness," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03881408, HAL.
    • Handle: RePEc:hal:cesptp:halshs-03881408
    DOI: 10.1007/s10107-021-01716-0
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    References listed on IDEAS

    as
    1. Camelia Bejan & Juan Camilo Gómez, 2012. "Using The Aspiration Core To Predict Coalition Formation," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 1-13.
    2. 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.
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    Cited by:

    1. Dylan Laplace Mermoud & Michel Grabisch & Peter Sudhölter, 2021. "Algorithmic aspects of core nonemptiness and core stability," Post-Print halshs-03354292, HAL.

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    More about this item

    Keywords

    Domination; Stable set; Core; TU game;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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