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Axiomatizations of the normalized Banzhaf value and the Shapley value

Author

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  • (*), 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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