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A Class of Consistent Share Functions for Games in Coalition Structure

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  • René van den Brink

    () (Tilburg University)

  • Gerard van der Laan

    () (Vrije Universiteit Amsterdam)

Abstract

A cooperative game with transferable utility describes a situation in which players can obtain certain payoffs by cooperation. A share function for such games is a function which assigns for every game a distribution of the payoffs over the players in the game. In this paper we consider cooperative games in which the players are organized into an a priori coalition structure being a finite partition of the set of players. We introduce a general method for defining a class of share functions for such games in coalition structure using a multiplication property that states that the share of a player in the total payoff is equal to its share in some internal game within its coalition multiplied by the share of this coalition in an external game between the coalitions. We show that these coalition structure share functions satisfy certain consistency properties. We provide axiomatizations of this class of coalition structure share functions 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. 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.
  2. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. René van den Brink & Gerard van der Laan, 1998. "The Normalized Banzhaf Value and the Banzhaf Share Function," Tinbergen Institute Discussion Papers 98-042/1, Tinbergen Institute.
  8. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  9. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. (*), 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.
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Citations

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Cited by:
  1. Gerard van der Laan & René van den Brink, 1998. "A Banzhaf Share Function for Cooperative Games in Coalition Structure," Tinbergen Institute Discussion Papers 98-043/1, Tinbergen Institute.
  2. 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.
  3. Kongo, Takumi, 2011. "Value of games with two-layered hypergraphs," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 114-119, September.
  4. Marcin Malawski, 2004. "‘‘Counting’’ power indices for games with a priori unions," Theory and Decision, Springer, vol. 56(1), pages 125-140, April.
  5. 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.

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