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Note On the core of a collection of coalitions

Author

Listed:
  • Hans Reijnierse

    (Department of Mathematics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands)

  • Jean Derks

    (Department of Mathematics, University of Maastricht, P.O.Box 616, 6200 MD Maastricht, The Netherlands)

Abstract

For a collection of subsets of a finite set N we define its core to be equal to the polyhedral cone {x∈IRN: ∑i∈N xi=0 and ∑i∈Sxi\geq0 for all S∈}. This note describes several applications of this concept in the field of cooperative game theory. Especially collections are considered with core equal to {0}. This property of a one-point core is proved to be equivalent to the non-degeneracy and balancedness of . Further, the notion of exact cover is discussed and used in a second characterization of collections with core equal to {0}.

Suggested Citation

  • Hans Reijnierse & Jean Derks, 1998. "Note On the core of a collection of coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 451-459.
    • Handle: RePEc:spr:jogath:v:27:y:1998:i:3:p:451-459
    Note: Received May 1997/Final version May 1998
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    Citations

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    Cited by:

    1. repec:hal:pseose:hal-00803233 is not listed on IDEAS
    2. Bochet, O.L.A. & Klaus, B.E., 2007. "A note on Dasgupta, Hammond, and Maskin's (1979) domain richness condition," Research Memorandum 039, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    3. Michel Grabisch, 2011. "Ensuring the boundedness of the core of games with restricted cooperation," Annals of Operations Research, Springer, vol. 191(1), pages 137-154, November.
    4. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 301-326, July.
    5. Csóka, Péter & Herings, P. Jean-Jacques & Kóczy, László Á., 2009. "Stable allocations of risk," Games and Economic Behavior, Elsevier, vol. 67(1), pages 266-276, September.
    6. Lijue Xie & Michel Grabisch, 2009. "The core of games on k-regular set systems," Post-Print halshs-00423922, HAL.
    7. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    8. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices: how to share benefits in a hierarchy," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00344802, HAL.
    9. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games," Documents de travail du Centre d'Economie de la Sorbonne 16081, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    10. Imma Curiel & Herbert Hamers & Stef Tijs & Jos Potters, 1997. "Restricted component additive games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(2), pages 213-220, June.
    11. Michel Grabisch & Lijue Xie, 2011. "The restricted core of games on distributive lattices: how to share benefits in a hierarchy," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 189-208, April.
    12. Péter Csóka & P. Herings & László Kóczy, 2011. "Balancedness conditions for exact games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(1), pages 41-52, August.
    13. 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.
    14. repec:hal:pseose:hal-01372858 is not listed on IDEAS
    15. Jesus Fco. Getan Olivan & Jesus Montes & Carlos Rafels Pallarola, 2006. "On the monotonic core," Working Papers in Economics 155, Universitat de Barcelona. Espai de Recerca en Economia.

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