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Axiomatizations of the normalized Banzhaf value and the Shapley value

Author

Listed:
  • (*), Gerard van der Laan

    (Department of Econometrics and Tinbergen Institute, Free University, De Boulelaan 1105, 1081 HV Amsterdam, The Netherlands)

  • RenÊ van den Brink

    (Department of Econometrics, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands)

Abstract

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. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways. An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the `grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the `grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms.

Suggested Citation

  • (*), 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.
  • Handle: RePEc:spr:sochwe:v:15:y:1998:i:4:p:567-582
    Note: Received: 10 April 1996 / Accepted: 2 June 1997
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    Citations

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    Cited by:

    1. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Theory and Decision, Springer, vol. 69(1), pages 69-96, July.
    2. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 309-319.
    3. 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.
    4. 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.
    5. Annick Laruelle, 1999. "- On The Choice Of A Power Index," Working Papers. Serie AD 1999-10, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    6. René van den Brink & Yukihiko Funaki & Yuan Ju, 2007. "Consistency, Monotonicity and Implementation of Egalitarian Shapley Values," Tinbergen Institute Discussion Papers 07-062/1, Tinbergen Institute.
    7. van den Brink, J.R. & van der Laan, G., 1999. "Potentials and Reduced Games for Share Functions," Discussion Paper 1999-41, Tilburg University, Center for Economic Research.
    8. van den Brink, Rene & van der Laan, Gerard, 2005. "A class of consistent share functions for games in coalition structure," Games and Economic Behavior, Elsevier, vol. 51(1), pages 193-212, April.
    9. Harald Wiese, 2012. "Values with exogenous payments," Theory and Decision, Springer, vol. 72(4), pages 485-508, April.
    10. Tobias Hiller, 2011. "A note on χ-values," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 58(4), pages 433-438, December.
    11. repec:hal:journl:halshs-00344457 is not listed on IDEAS
    12. Yan-An Hwang & Yu-Hsien Liao, 2010. "Consistency and dynamic approach of indexes," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(4), pages 679-694, April.
    13. Palestini, Arsen & Pignataro, Giuseppe, 2016. "A graph-based approach to inequality assessment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 455(C), pages 65-78.
    14. Bernardi Mauro & Roy Cerqueti & Arsen Palestini, 2016. "Allocation of risk capital in a cost cooperative game induced by a modified Expected Shortfall," Papers 1608.02365, arXiv.org.
    15. Oishi, Takayuki & Nakayama, Mikio & Hokari, Toru & Funaki, Yukihiko, 2016. "Duality and anti-duality in TU games applied to solutions, axioms, and axiomatizations," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 44-53.
    16. van den Brink, J.R. & Ruys, P.H.M. & Semenov, R., 1999. "Governance of Clubs and Firms with Cultural Dimensions," Discussion Paper 1999-101, Tilburg University, Center for Economic Research.
    17. repec:spr:sochwe:v:50:y:2018:i:1:d:10.1007_s00355-017-1074-4 is not listed on IDEAS
    18. Barua, Rana & Chakravarty, Satya R. & Sarkar, Palash, 2009. "Minimal-axiom characterizations of the Coleman and Banzhaf indices of voting power," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 367-375, November.
    19. 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.
    20. Rene van den Brink, 2009. "Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games," Tinbergen Institute Discussion Papers 09-065/1, Tinbergen Institute.

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