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On the Replication of the Pre-Kernel and Related Solutions

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  • Meinhardt, Holger Ingmar

Abstract

Based on results discussed by Meinhardt (2013), which presents a dual characterization of the prekernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the single-valuedness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. Hence, a bargaining outcome related to this pre-kernel element is stable. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. In addition, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point. Finally, we apply the same techniques to related solutions of the pre-kernel, namely the modiclus and anti-pre-kernel, to work out replication results for them.

Suggested Citation

  • Meinhardt, Holger Ingmar, 2020. "On the Replication of the Pre-Kernel and Related Solutions," MPRA Paper 102676, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:102676
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    File URL: https://mpra.ub.uni-muenchen.de/102676/1/MPRA_paper_102676.pdf
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    References listed on IDEAS

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    2. Driessen, Theo S.H. & Meinhardt, Holger I., 2005. "Convexity of oligopoly games without transferable technologies," Mathematical Social Sciences, Elsevier, vol. 50(1), pages 102-126, July.
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    7. Meinhardt, Holger Ingmar, 2014. "A Note on the Computation of the Pre-Kernel for Permutation Games," MPRA Paper 59365, University Library of Munich, Germany.
    8. T. Raghavan & Peter Sudhölter, 2005. "The modiclus and core stability," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(4), pages 467-478, November.
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    10. Holger Ingmar Meinhardt, 2014. "The Pre-Kernel as a Tractable Solution for Cooperative Games," Theory and Decision Library C, Springer, edition 127, number 978-3-642-39549-9, December.
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    Cited by:

    1. Meinhardt, Holger Ingmar, 2021. "Disentangle the Florentine Families Network by the Pre-Kernel," MPRA Paper 106482, University Library of Munich, Germany.

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    More about this item

    Keywords

    Transferable Utility Game; Pre-Kernel; Pre-Nucleolus; Anti-Pre-Nucleolus; Modiclus; Uniqueness of the Pre-Kernel; Convex Analysis; Fenchel-Moreau Conjugation; Indirect Function; Stability Analysis.;
    All these keywords.

    JEL classification:

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

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