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A class of consistent share functions for games in coalition structure

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

Abstract

A cooperative game with transferable utility -or simply a TU-game- describes a situation in which players can obtain certain payoffs by cooperation.A value function for these games is a function which assigns to every such a game a distribution of the payoffs over the players in the game.An alternative type of solutions are share functions which assign to every player in a TU-game its share in the payoffs to be distributed.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 player i in the total payoff is equal to the share of player i in some internal game within i 's a priori coalition, multiplied by the share of this coalition in an external game between the a priori given 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

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

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Web page: http://www.elsevier.com/locate/inca/622836

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References

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  1. 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.
  2. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  3. 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.
  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. 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.
  6. 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.
  7. 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.
  8. 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.
  9. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  10. 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.
  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. (*), 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.
  13. 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.
  14. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  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. 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.
  2. 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.
  3. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00803233, HAL.
  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. 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.
  6. 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.
  7. Nicola G. Andjiga & Sebastien Courtin, 2013. "Coalition configurations and share functions," Working Papers hal-00914883, HAL.
  8. 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.
  9. 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.
  10. Kongo, Takumi, 2011. "Value of games with two-layered hypergraphs," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 114-119, September.
  11. 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.

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