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A Banzhaf share function for cooperative games in coalition structure


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  • Gerard van der Laan


  • René van den Brink


A cooperative game with transferable utilities, or simply a TU-game, describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Well-known solution concepts for TU-games are the Shapley value and the Banzhaf value. The Shapley value is efficient, i.e. the total payoff is equal to the worth of the `grand coalition', but the Banzhaf value is not efficient. An alternative type of solution is the concept of share functions, being functions which assign to every player in a TU-game its share in the worth of the grand coalition. The Shapley (respectively Banzhaf) share function is the share function giving to each player his Shapley (Banzhaf) value divided by the sum of the Shapley (Banzhaf) values over all players. In this paper we consider cooperative games in which the players are organized into a coalition structure being a finite partition of the set of players. A value function for games in coalition structure has been proposed by Owen. The Owen value can be considered as a direct generalization of the Shapley value to games in coalition structure. We define the Owen share function as the share function for games in coalition structure giving to each player his Owen value divided by the sum of the Owen values over all players. We then show that this Owen share function satisfies a multiplicity property, namely that the Owen share of a player i in a coalition K is equal to the Shapley share of coalition K in a first level game between the coalitions in the coalition structure times the Shapley share of player i in a second level game between the players in K. We show that analogously a Banzhaf share function for games with coalition structure can be obtained by defining the share of a player in some coalition as the Banzhaf share of the coalition in a first level game between the coalitions times the Banzha

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Bibliographic Info

Article provided by Springer in its journal Theory and Decision.

Volume (Year): 53 (2002)
Issue (Month): 1 (August)
Pages: 61-86

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Handle: RePEc:kap:theord:v:53:y:2002:i:1:p:61-86

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Keywords: TU-game; coalition structure; Banzhaf share function; multiplication property; consistency;

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  1. René van den Brink & Gerard van der Laan, 2001. "A Class of Consistent Share Functions for Games in Coalition Structure," Tinbergen Institute Discussion Papers 01-044/1, Tinbergen Institute.
  2. Hart, Sergiu & Kurz, Mordecai, 1983. "Endogenous Formation of Coalitions," Econometrica, Econometric Society, vol. 51(4), pages 1047-64, July.
  3. 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.
  4. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  5. 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.
  6. (*), 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.
  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. Owen, Guillermo & Winter, Eyal, 1992. "The multilinear extension and the coalition structure value," Games and Economic Behavior, Elsevier, vol. 4(4), pages 582-587, October.
  9. 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.
  10. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  11. 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.
  12. 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.
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Cited by:
  1. Brink, J.R. van den & Laan, G. van der, 2001. "A Class of Consistent Share Functions For Games in Coalition Structure," Discussion Paper 2001-33, Tilburg University, Center for Economic Research.
  2. 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.
  3. 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.
  4. Nicola G. Andjiga & Sebastien Courtin, 2013. "Coalition configurations and share functions," Working Papers hal-00914883, HAL.
  5. Kongo, Takumi, 2011. "Value of games with two-layered hypergraphs," Mathematical Social Sciences, Elsevier, vol. 62(2), pages 114-119, September.
  6. repec:hal:cesptp:halshs-00308741 is not listed on IDEAS
  7. repec:dgr:uvatin:2012052 is not listed on IDEAS
  8. repec:hal:journl:halshs-00308741 is not listed on IDEAS
  9. repec:hal:journl:halshs-00344457 is not listed on IDEAS


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