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A class of consistent share functions for games in coalition structure

  • van den Brink, Rene
  • van der Laan, Gerard

This discussion paper led to a publication in 'Games and Economic Behavior', 2005, 51, 193-212. A cooperative game with transferable utility describes a situation in which players can obtain certain payoffs by cooperation. A sharefunction 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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Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 51 (2005)
Issue (Month): 1 (April)
Pages: 193-212

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Handle: RePEc:eee:gamebe:v:51:y:2005:i:1:p:193-212
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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  1. van der Laan, G. & van den Brink, J.R., 1998. "A Banzhaf share function for cooperative games in coalition structure," Discussion Paper 1998-66, Tilburg University, Center for Economic Research.
  2. Andrzej S. Nowak, 1997. "note: On an Axiomatization of the Banzhaf Value without the Additivity Axiom," International Journal of Game Theory, Springer;Game Theory Society, vol. 26(1), pages 137-141.
  3. (*), Gerard van der Laan & RenÊ van den Brink, 1998. "Axiomatizations of the normalized Banzhaf value and the Shapley value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(4), pages 567-582.
  4. Winter, Eyal, 1989. "A Value for Cooperative Games with Levels Structure of Cooperation," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 227-40.
  5. Gerard van der Laan & René van den Brink, 2001. "Core concepts for share vectors," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(4), pages 759-784.
  6. 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.
  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. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
  9. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
  10. 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.
  11. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-81.
  12. José Alonso-Meijide & M. Fiestras-Janeiro, 2002. "Modification of the Banzhaf Value for Games with a Coalition Structure," Annals of Operations Research, Springer, vol. 109(1), pages 213-227, January.
  13. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  14. 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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