Abstract
In this paper we generalize the graph Banzhaf value, proposed by Alonso-Meijide and Fiestras-Janeiro (Naval Res Logist 53(3):198–203, 2006) in the deterministic communication situations, to the generalized probabilistic communication situations. This new value is called the probabilistic Banzhaf value. We provide two axiomatic characterizations of the value by the probabilistic versions of component total power, fairness and balanced contributions. Furthermore, we give an alternative characterization of the value by using the probabilistic player potential function.
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References
Alonso-Meijide, J. M., Carreras, F., Fiestras-Janeiro, M. G., & Owen, G. (2007). A comparative axiomatic characterization of the Banzhaf–Owen coalitional value. Decision Support Systems, 43, 701–712.
Alonso-Meijide, J. M., & Fiestras-Janeiro, M. G. (2002). Modification of the Banzhaf value for games with a coalition structure. Annals of Operations Research, 109, 213–227.
Alonso-Meijide, J. M., & Fiestras-Janeiro, M. G. (2006). The Banzhaf value and communication situations. Naval Research Logistics, 53(3), 198–203.
Amer, R., Carreras, F., & Giménez, J. M. (2002). The modified Banzhaf value for games with coalition structure: An axiomatic characterization. Mathematical Social Sciences, 43, 45–54.
Banzhaf, J. F. (1965). Weighted voting does not work: A mathematical analysis. Rutgers Law Review, 19, 317–343.
Calvo, E., Lasaga, J., & van den Nouweland, A. (1999). Values of games with probabilistic graphs. Mathematical Social Sciences, 37, 79–95.
Casajus, A. (2011). Marginality, differential marginality, and the Banzhaf value. Theory and Decision, 71, 365–372.
Dragan, I. (1996). New mathematical properties of the Banzhaf value. European Journal of Operational Research, 95(2), 451–463.
Feltkamp, V. (1995). Alternative axiomatic characterization of the Shapley and Banzhaf values. International Journal of Game Theory, 24, 179–186.
Ghintran, A., González-Arangüena, E., & Manuel, C. (2012). A probabilistic position value. Annals of Operations Research, 1, 183–196.
Gómez, D., González-Arangüena, E., Manuel, C., & Owen, G. (2008). A value for generalized probabilistic communication situations. European Journal of Operational Research, 190, 539–556.
Haimanko, O. (2019). Composition independence in compound games: A characterization of the Banzhaf power index and the Banzhaf value. International Journal of Game Theory, 48, 755–768.
Harsanyi, J. C. (1959). A bargaining model for cooperative \(n\)-person games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games IV (pp. 325–355). Princeton, NJ: Princeton University Press.
Meessen, R. (1988). Communication games, Master’s thesis, Department of Mathematics, University of Nijmegen, The Netherlands (in Dutch).
Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2(3), 225–229.
Owen, G. (1975). Multilinear extensions and the Banzhaf value. Naval Research Logistics, 22, 741–750.
Owen, G. (1986). Values of graph-restricted games. SIAM Journal on Discrete Mathematics, 7, 210–220.
Ridaoui, M., Grabisch, M., & Labreuche, C. (2018). An axiomatization of the Banzhaf value and interaction index for multichoice games. Université Panthéon-Sorbonne (Paris 1), Centre d’Economie de la Sorbonne.
Shapley, L. S. (1953). A value for \(n\)-person games. In H. Kuhn & A. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press.
Slikker, M. (2005). Link monotonic allocation schemes. International Game Theory Review, 7, 473–489.
Slikker, M., van den Nouweland, A. (2001). Social and economic networks in cooperative game theory. In Theory and Decision Library (Vol. 27).
Acknowledgements
The authors are grateful to anonymous referees for valuable comments and suggestions that contributed greatly to the improvement of this paper. This research was partially supported by the National Natural Science Foundation of China (No. 11971298), Zhejiang Federation of Humanities and Social Sciences Circles Research Project (No. 2018B08), Ningbo Soft Science Research Project (No. 2017A10005) and the National Social Science Fund of China (No. 19BGL001).
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Shi, J., Shan, E. The Banzhaf value for generalized probabilistic communication situations. Ann Oper Res 301, 225–244 (2021). https://doi.org/10.1007/s10479-020-03914-z
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DOI: https://doi.org/10.1007/s10479-020-03914-z