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The prenucleolus and the prekernel for games with communication structures

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  • Anna Khmelnitskaya
  • Peter Sudhölter

Abstract

It is well-known that the prekernel on the class of TU games is uniquely determined by non-emptiness, Pareto efficiency (EFF), covariance under strategic equivalence (COV), the equal treatment property, the reduced game property (RGP), and its converse. We show that the prekernel on the class of TU games restricted to the connected coalitions with respect to communication structures may be axiomatized by suitably generalized axioms. Moreover, it is shown that the prenucleolus, the unique solution concept on the class of TU games that satisfies singlevaluedness, COV, anonymity, and RGP, may be characterized by suitably generalized versions of these axioms together with a property that is called “independence of irrelevant connections”. This property requires that any element of the solution to a game with communication structure is an element of the solution to the game that allows unrestricted cooperation in all connected components, provided that each newly connected coalition is sufficiently charged, i.e., receives a sufficiently small worth. Both characterization results may be extended to games with conference structures. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Anna Khmelnitskaya & Peter Sudhölter, 2013. "The prenucleolus and the prekernel for games with communication structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 285-299, October.
    • Handle: RePEc:spr:mathme:v:78:y:2013:i:2:p:285-299
    DOI: 10.1007/s00186-013-0444-7
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    References listed on IDEAS

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    1. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z., 2010. "The average tree solution for cooperative games with communication structure," Games and Economic Behavior, Elsevier, vol. 68(2), pages 626-633, March.
    2. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Katsev, Ilya & Yanovskaya, Elena, 2013. "The prenucleolus for games with restricted cooperation," Mathematical Social Sciences, Elsevier, vol. 66(1), pages 56-65.
    4. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 187-200.
    5. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    6. Guni Orshan & Peter Sudhölter, 2010. "The positive core of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 113-136, March.
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    Cited by:

    1. Schouten, Jop & Dietzenbacher, Bas & Borm, Peter, 2019. "The Nucleolus and Inheritance of Properties in Communication Situations," Discussion Paper 2019-008, Tilburg University, Center for Economic Research.
    2. Elena Iñarra & Roberto Serrano & Ken-Ichi Shimomura, 2020. "The Nucleolus, the Kernel, and the Bargaining Set: An Update," Revue économique, Presses de Sciences-Po, vol. 71(2), pages 225-266.
    3. M. Albizuri & Peter Sudhölter, 2016. "Characterizations of the core of TU and NTU games with communication structures," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(2), pages 451-475, February.
    4. J. Schouten & B. Dietzenbacher & P. Borm, 2022. "The nucleolus and inheritance of properties in communication situations," Annals of Operations Research, Springer, vol. 318(2), pages 1117-1135, November.
    5. Michel Grabisch & Hervé Moulin & José Manuel Zarzuelo, 2024. "Professor Peter Sudhölter (1957–2024)," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 289-294, June.
    6. Schouten, Jop, 2022. "Cooperation, allocation and strategy in interactive decision-making," Other publications TiSEM d5d41448-8033-4f6b-8ec0-c, Tilburg University, School of Economics and Management.

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    More about this item

    Keywords

    TU game; Solution concept; Communication and conference structure; Nucleolus; Kernel; C71;
    All these keywords.

    JEL classification:

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

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