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The Banzhaf Value in the Presence of Externalities

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Abstract

We propose two generalizations of the Banzhaf value for partition function form games. In both cases, our approach is based on probability distributions over the set of possible coalition structures that may arise for any given set of agents. First, we introduce a family of values, one for each collection of the latter probability distributions, defined as the Banzhaf value of an expected coalitional game. Then, we provide two characterization results for this new family of values within the framework of all partition function games. Both results rely on a property of neutrality with respect to am algamation of players. Second, as this collusion transformation fails to be meanin gful for simple games in partition function form, we propose another generalization of the Banzhaf value which also builds on probability distributions of the above type. This latter family is characterized by means of a neutrality property which uses an amalgamation transformation of players for which simple games are close

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  • Mikel Álvarez-Mozos & Oriol Tejada Pinyol, 2014. "The Banzhaf Value in the Presence of Externalities," UB Economics Working Papers 2014/302, Universitat de Barcelona, Facultat d'Economia i Empresa, UB Economics.
  • Handle: RePEc:ewp:wpaper:302web
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    References listed on IDEAS

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    1. Edward M. Bolger, 2002. "Characterizations of two power indices for voting games with r alternatives," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 709-721.
    2. Macho-Stadler, Ines & Perez-Castrillo, David & Wettstein, David, 2007. "Sharing the surplus: An extension of the Shapley value for environments with externalities," Journal of Economic Theory, Elsevier, vol. 135(1), pages 339-356, July.
    3. Dutta, Bhaskar & Ehlers, Lars & Kar, Anirban, 2010. "Externalities, potential, value and consistency," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2380-2411, November.
    4. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    5. Marcin Malawski, 2002. "Equal treatment, symmetry and Banzhaf value axiomatizations," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 47-67.
    6. Josep Freixas & William S. Zwicker, 2003. "Weighted voting, abstention, and multiple levels of approval," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 399-431, December.
    7. Geoffroy de Clippel & Roberto Serrano, 2008. "Marginal Contributions and Externalities in the Value," Econometrica, Econometric Society, vol. 76(6), pages 1413-1436, November.
    8. 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.
    9. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    10. Hafalir, Isa E., 2007. "Efficiency in coalition games with externalities," Games and Economic Behavior, Elsevier, vol. 61(2), pages 242-258, November.
    11. André Casajus, 2012. "Amalgamating players, symmetry, and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 497-515, August.
    12. Bolger, E M, 1986. "Power Indices for Multicandidate Voting Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 175-186.
    13. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
    14. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
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    Cited by:

    1. José María Alonso-Meijide & Mikel Álvarez-Mozos & María Gloria Fiestras-Janeiro, 2015. "Power Indices and Minimal Winning Coalitions in Simple Games with Externalities Abstract: We propose a generalization of simple games to situations with coalitional externalities. The main novelty of ," UB Economics Working Papers 2015/328, Universitat de Barcelona, Facultat d'Economia i Empresa, UB Economics.
    2. repec:spr:grdene:v:26:y:2017:i:6:d:10.1007_s10726-017-9542-x is not listed on IDEAS

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    Keywords

    Banzhaf value; Externalities; Games in partition function form; Simplegames.;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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