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The Banzhaf value in the presence of externalities

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  • M. Álvarez-Mozos
  • O. Tejada

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 coalition structures that may arise for any given set of players. First, we introduce a family of values, one for each collection of these latter probability distributions, defined as the Banzhaf value of a coalitional game obtained as the expectation taken according to the given probability distributions of the original partition function form game. For each value of the family we provide two characterization results within the set of all partition function form games. Both results rely on a property of neutrality with respect to the amalgamation of players. Second, we propose another family of values that differ from the previous ones in that the latter values take into account only the information about the most likely coalition structure that may arise according to the given probability distributions. Each value of the second family is also characterized in two results by means of a collusion neutrality property. Unlike the characterizations of the first approach, these characterizations can be restricted to the set of simple games in partition function form. Copyright Springer-Verlag Berlin Heidelberg 2015

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  • M. Álvarez-Mozos & O. Tejada, 2015. "The Banzhaf value in the presence of externalities," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(4), pages 781-805, April.
    • Handle: RePEc:spr:sochwe:v:44:y:2015:i:4:p:781-805
    DOI: 10.1007/s00355-014-0861-4
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    1. Dutta, Bhaskar & Ehlers, Lars & Kar, Anirban, 2010. "Externalities, potential, value and consistency," Journal of Economic Theory, Elsevier, vol. 145(6), pages 2380-2411, November.
    2. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    3. 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.
    4. R. M. Thrall & W. F. Lucas, 1963. "N‐person games in partition function form," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 10(1), pages 281-298, March.
    5. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    6. 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.
    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. M. J. Albizuri & J. Arin & J. Rubio, 2005. "An Axiom System For A Value For Games In Partition Function Form," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 63-72.
    9. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Kim Hang Pham Do & Henk Norde, 2007. "The Shapley Value For Partition Function Form Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 353-360.
    15. Hafalir, Isa E., 2007. "Efficiency in coalition games with externalities," Games and Economic Behavior, Elsevier, vol. 61(2), pages 242-258, November.
    16. 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.
    17. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
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    1. J. M. Alonso-Meijide & M. Álvarez-Mozos & M. G. Fiestras-Janeiro, 2017. "Power Indices and Minimal Winning Coalitions for Simple Games in Partition Function Form," Group Decision and Negotiation, Springer, vol. 26(6), pages 1231-1245, November.
    2. László Á. Kóczy, 2018. "Partition Function Form Games," Theory and Decision Library C, Springer, number 978-3-319-69841-0, March.
    3. 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 School of Economics Working Papers 2015/328, University of Barcelona School of Economics.
    4. G. Arévalo-Iglesias & M. Álvarez-Mozos, 2020. "Power distribution in the Basque Parliament using games with externalities," Theory and Decision, Springer, vol. 89(2), pages 157-178, September.

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    JEL classification:

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

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