Kybernetika 55 no. 1, 63-80, 2019

A bi-average tree solution for probabilistic communication situations with fuzzy coalition

Xianghui Li, Hao Sun and Dongshuang HouDOI: 10.14736/kyb-2019-1-0063

Abstract:

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.

Keywords:

fuzzy coalition, probabilistic communication situation, average tree solution, maximal product spanning tree

Classification:

05C57, 05C72, 91A12

paper.pdf

References:

  1. J. P. Aubin: Coeur et valeur des jeux flous à paiements latéraux. Comptes Rendus Hebdomadaires des Séances de 1'Académie des Sciences 279-A (1974), 891-894.   CrossRef
  2. K. R. Bhutani and A. Rosenfeld: Strong arcs in fuzzy graphs. Inform. Sci. 152 (2003), 319-322.   DOI:10.1016/s0020-0255(02)00411-5
  3. P. Borm, G. Owen and S. Tijs: On the position value for communication situations. SIAM J. Discrete Math. 5 (1992), 305-320.   DOI:10.1137/0405023
  4. D. Butnariu: Stability and Shapley value for an n-persons fuzzy game. Fuzzy Sets and Systems 4 (1980), 63-72.   DOI:10.1016/0165-0114(80)90064-0
  5. E. Calvo, J. Lasaga and A. van den Nouweland: Values of games with probabilistic graphs. Math. Social Sci. 37 (1999), 79-95.   DOI:10.1016/s0165-4896(98)00013-4
  6. J. M. Gallardo, N. Jiménez, A. Jiménez-Losada and E. Lebrón: Games with fuzzy authorization structure: A Shapley value. Fuzzy Sets and Systems 272 (2015), 115-125.   DOI:10.1016/j.fss.2014.09.002
  7. D. Gómez, E. González-Arangüena, C. Manuel and G. Owen: A value for generalized probabilistic communication situations. Europ. J. Oper. Res. 190 (2008), 539-556.   DOI:10.1016/j.ejor.2007.06.040
  8. P. J. J. Herings, G. van der Laan and D. Talman: The average tree solution for cycle-free graph games. Games and Economic Behavior 62 (2008), 77-92.   DOI:10.1016/j.geb.2007.03.007
  9. A. Jiménez-Losada, J. R. Fernández, M. Ordóñez and M. Grabisch: Games on fuzzy communication structures with Choquet players. Europ. J. Oper. Res. 207 (2010), 836-847.   DOI:10.1016/j.ejor.2010.06.014
  10. X. Li, H. Sun and D Hou: On the position value for communication situations with fuzzy coalition. J. Intell. Fuzzy Systems 33 (2017), 113-124.   DOI:10.3233/jifs-16117
  11. R. B. Myerson: Graphs and cooperation in games. Mathematics of Operations Research 2 (1977), 225-229.   DOI:10.1287/moor.2.3.225
  12. M. Tsurumi, T. Tanino and M. Inuiguchi: A Shapley function on a class of cooperative fuzzy games. Europ. J. Oper. Res. 129 (2001), 596-618.   DOI:10.1016/s0377-2217(99)00471-3
  13. X. Yu and Q. Zhang: The fuzzy core in games with fuzzy coalitions. J. Computat. Appl. Math. 230 (2009), 173-186.   DOI:10.1016/j.cam.2008.11.004
  14. G. Xu, X. Li, H. Sun and J. Su: The Myerson value for cooperative games on communication structure with fuzzy coalition. J. Intell. Fuzzy Systems 33 (2017), 27-39.   DOI:10.3233/jifs-16080