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A Class of Consistent Share Functions for Games in Coalition Structure

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Author Info

  • René van den Brink

    ()
    (Tilburg University)

  • Gerard van der Laan

    ()
    (Vrije Universiteit Amsterdam)

Abstract

A cooperative game with transferable utility describes a situation inwhich players can obtain certain payoffs by cooperation. A sharefunction for such games is a function which assigns for every game adistribution of the payoffs over the players in the game.In this paper we consider cooperative games in which the playersare organized into an a priori coalition structure being a finitepartition of the set of players. We introduce a general method fordefining a class of share functions for such games in coalitionstructure using a multiplication property that states that theshare of a player in the total payoff is equal to its share in someinternal game within its coalition multiplied by the share of thiscoalition in an external game between the coalitions. We show thatthese coalitionstructure share functions satisfy certain consistency properties. Weprovide axiomatizations of this class of coalition structure sharefunctions using these consistency and multiplication properties.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 01-044/1.

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Date of creation: 27 Apr 2001
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Handle: RePEc:dgr:uvatin:20010044

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Web page: http://www.tinbergen.nl

Related research

Keywords: TU-Game; coalition structure; share function; multiplication property; consistency;

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References

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  1. Herings,P. Jean-Jacques & Laan,Gerard,van der & Talman,Dolf, 2002. "Cooperative Games in Graph Structure," Research Memorandum 011, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  2. Brink, J.R. van den & Laan, G. van der, 1999. "Core Concepts for Share Vectors," Discussion Paper 1999-64, Tilburg University, Center for Economic Research.
  3. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  4. Brink, J.R. van den & Laan, G. van der, 1998. "The normalized Banzhaf value and the Banzhaf share function," Research Memorandum 764, Tilburg University, Faculty of Economics and Business Administration.
  5. Winter, Eyal, 1989. "A Value for Cooperative Games with Levels Structure of Cooperation," International Journal of Game Theory, Springer, vol. 18(2), pages 227-40.
  6. Laan, G. van der & Brink, J.R. van den, 1998. "A Banzhaf share function for cooperative games in coalition structure," Discussion Paper 1998-66, Tilburg University, Center for Economic Research.
  7. Pekec, Aleksandar, 2001. "Meaningful and meaningless solutions for cooperative n-person games," European Journal of Operational Research, Elsevier, vol. 133(3), pages 608-623, September.
  8. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  9. (*), Gerard van der Laan & RenÊ van den Brink, 1998. "Axiomatizations of the normalized Banzhaf value and the Shapley value," Social Choice and Welfare, Springer, vol. 15(4), pages 567-582.
  10. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  11. Winter, Eyal, 1992. "The consistency and potential for values of games with coalition structure," Games and Economic Behavior, Elsevier, vol. 4(1), pages 132-144, January.
  12. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer, vol. 28(4), pages 547-565.
  13. Andrzej S. Nowak, 1997. "note: On an Axiomatization of the Banzhaf Value without the Additivity Axiom," International Journal of Game Theory, Springer, vol. 26(1), pages 137-141.
  14. Gerard van der Laan & René van den Brink, 1998. "Axiomatization of a class of share functions for n-person games," Theory and Decision, Springer, vol. 44(2), pages 117-148, April.
  15. Amer, Rafael & Carreras, Francese & Gimenez, Jose Miguel, 2002. "The modified Banzhaf value for games with coalition structure: an axiomatic characterization," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 45-54, January.
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Citations

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Cited by:
  1. Sébastien Courtin, 2011. "Power in the European Union: an evaluation according to a priori relations between states," Economics Bulletin, AccessEcon, vol. 31(1), pages 534-545.
  2. Nicola G. Andjiga & Sebastien Courtin, 2013. "Coalition configurations and share functions," Working Papers hal-00914883, HAL.
  3. Yoshio Kamijo, 2013. "The collective value: a new solution for games with coalition structures," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 21(3), pages 572-589, October.
  4. Laan, G. van der & Brink, J.R. van den, 1998. "A Banzhaf share function for cooperative games in coalition structure," Discussion Paper 1998-66, Tilburg University, Center for Economic Research.
  5. Rene van den Brink & Anna Khmelnitskaya & Gerard van der Laan, 2011. "An Owen-Type Value for Games with Two-Level Communication Structures," Tinbergen Institute Discussion Papers 11-089/1, Tinbergen Institute.
  6. Marcin Malawski, 2004. "‘‘Counting’’ power indices for games with a priori unions," Theory and Decision, Springer, vol. 56(1), pages 125-140, April.
  7. Álvarez-Mozos, M. & van den Brink, R. & van der Laan, G. & Tejada, O., 2013. "Share functions for cooperative games with levels structure of cooperation," European Journal of Operational Research, Elsevier, vol. 224(1), pages 167-179.
  8. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00803233, HAL.
  9. Kongo, Takumi, 2011. "Value of games with two-layered hypergraphs," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 114-119, September.
  10. Mikel Alvarez-Mozos & Rene van den Brink & Gerard van der Laan & Oriol Tejada, 2012. "Share Functions for Cooperative Games with Levels Structure of Cooperation," Tinbergen Institute Discussion Papers 12-052/1, Tinbergen Institute.
  11. Rene van den Brink & Gerard van der Laan & Nigel Moes, 2011. "Two Values for Transferable Utility Games with Coalition and Graph Structure," Tinbergen Institute Discussion Papers 11-164/1, Tinbergen Institute.

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