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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. 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.
  2. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  3. 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.
  4. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2000. "Cooperative Games in Graph Structure," Discussion Paper 2000-90, Tilburg University, Center for Economic Research.
  5. 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.
  6. 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.
  7. Eyal Winter, 1989. "The Consistency and Potential for Values of Games with Coalition Structure," Discussion Paper Serie A 242, University of Bonn, Germany.
  8. 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.
  9. 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.
  10. (*), 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.
  11. 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.
  12. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  13. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  14. Gerard van der Laan & René van den Brink, 2001. "Core concepts for share vectors," Social Choice and Welfare, Springer, vol. 18(4), pages 759-784.
  15. 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.
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Citations

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Cited by:
  1. 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.
  2. 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.
  3. Marcin Malawski, 2004. "‘‘Counting’’ power indices for games with a priori unions," Theory and Decision, Springer, vol. 56(1), pages 125-140, April.
  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. Michel Grabisch, 2009. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00445171, HAL.
  6. 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.
  7. Nicola G. Andjiga & Sebastien Courtin, 2013. "Coalition configurations and share functions," Working Papers hal-00914883, HAL.
  8. 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.
  9. 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.
  10. 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.
  11. Kongo, Takumi, 2011. "Value of games with two-layered hypergraphs," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 114-119, September.

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