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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. 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.
    2. 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.
    3. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. repec:spr:compst:v:74:y:2011:i:1:p:41-52 is not listed on IDEAS
    10. 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.
    11. repec:hal:journl:halshs-00423922 is not listed on IDEAS
    12. repec:hal:journl:halshs-00445171 is not listed on IDEAS
    13. repec:hal:cesptp:hal-00803233 is not listed on IDEAS
    14. repec:spr:compst:v:73:y:2011:i:2:p:189-208 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.
    16. repec:spr:compst:v:45:y:1997:i:2:p:213-220 is not listed on IDEAS
    17. repec:hal:journl:halshs-00344802 is not listed on IDEAS

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