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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. Eyal Winter, 1989. "The Consistency and Potential for Values of Games with Coalition Structure," Discussion Paper Serie A 242, University of Bonn, Germany.
  2. 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.
  3. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
  4. 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.
  5. 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.
  6. van den Brink, J.R. & van der Laan, G., 1999. "Core Concepts for Share Vectors," Discussion Paper 1999-64, Tilburg University, Center for Economic Research.
  7. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  8. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-81.
  9. 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.
  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;The Society for Social Choice and Welfare, vol. 15(4), pages 567-582.
  11. Grabisch, M. & Roubens, M., 1998. "An Axiomatic Approach to the Concept of Interaction Among Players in Cooperative Games," Liege - Groupe d'Etude des Mathematiques du Management et de l'Economie 9818, UNIVERSITE DE LIEGE, Faculte d'economie, de gestion et de sciences sociales, Groupe d'Etude des Mathematiques du Management et de l'Economie.
  12. Gerard van der Laan & René van den Brink, 2002. "A Banzhaf share function for cooperative games in coalition structure," Theory and Decision, Springer, vol. 53(1), pages 61-86, August.
  13. 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.
  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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