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A bi-average tree solution for probabilistic communication situations with fuzzy coalition. (English) Zbl 1449.05187
Summary: A probabilistic communication structure considers the setting with communication restrictions in which each pair of players has a probability to communicate directly. In this paper, we consider a more general framework, called a probabilistic communication structure with fuzzy coalition, that allows any player to have a participation degree to cooperate within a coalition. A maximal product spanning tree, indicating a way of the greatest possibility to communicate among the players, is introduced, where the unique path from one player to another is optimal. We present a feasible procedure to find the maximal product spanning trees. Furthermore, for games under this model, a new solution concept in terms of the average tree solution is proposed and axiomatized by defining a restricted game in Choquet integral form.
##### MSC:
 05C57 Games on graphs (graph-theoretic aspects) 05C72 Fractional graph theory, fuzzy graph theory 91A12 Cooperative games 05C76 Graph operations (line graphs, products, etc.)
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##### References:
 [1] Aubin, J. P., Coeur et valeur des jeux flous à paiements latéraux., Comptes Rendus Hebdomadaires des Séances de 1’Académie des Sciences 279-A (1974), 891-894 [2] Bhutani, K. R.; Rosenfeld, A., Strong arcs in fuzzy graphs., Inform. Sci. 152 (2003), 319-322 [3] Borm, P.; Owen, G.; Tijs, S., On the position value for communication situations., SIAM J. Discrete Math. 5 (1992), 305-320 [4] Butnariu, D., Stability and Shapley value for an n-persons fuzzy game., Fuzzy Sets and Systems 4 (1980), 63-72 [5] Calvo, E.; Lasaga, J.; Nouweland, A. van den, Values of games with probabilistic graphs., Math. Social Sci. 37 (1999), 79-95 [6] Gallardo, J. M.; Jiménez, N.; Jiménez-Losada, A.; Lebrón, E., Games with fuzzy authorization structure: A Shapley value., Fuzzy Sets and Systems 272 (2015), 115-125 [7] Gómez, D.; González-Arangüena, E.; Manuel, C.; Owen, G., A value for generalized probabilistic communication situations., Europ. J. Oper. Res. 190 (2008), 539-556 [8] Herings, P. J. J.; Laan, G. van der; Talman, D., The average tree solution for cycle-free graph games., Games and Economic Behavior 62 (2008), 77-92 [9] Jiménez-Losada, A.; Fernández, J. R.; Ordóñez, M.; Grabisch, M., Games on fuzzy communication structures with Choquet players., Europ. J. Oper. Res. 207 (2010), 836-847 [10] Li, X.; Sun, H.; Hou, D., On the position value for communication situations with fuzzy coalition., J. Intell. Fuzzy Systems 33 (2017), 113-124 [11] Myerson, R. B., Graphs and cooperation in games., Mathematics of Operations Research 2 (1977), 225-229 [12] Tsurumi, M.; Tanino, T.; Inuiguchi, M., A Shapley function on a class of cooperative fuzzy games., Europ. J. Oper. Res. 129 (2001), 596-618 [13] Yu, X.; Zhang, Q., The fuzzy core in games with fuzzy coalitions., J. Computat. Appl. Math. 230 (2009), 173-186 [14] Xu, G.; Li, X.; Sun, H.; Su, J., The Myerson value for cooperative games on communication structure with fuzzy coalition., J. Intell. Fuzzy Systems 33 (2017), 27-39
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