American Institute of Mathematical Sciences

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August & September  2019, 12(4&5): 1187-1198. doi: 10.3934/dcdss.2019082

Cores and optimal fuzzy communication structures of fuzzy games

Jiaquan Zhan 1,, and  Fanyong Meng 2,

1. 

School of Management, Qingdao University of Technology, Qingdao 266520, China
2. 

Business School, Central South University, Changsha 410083, China

* Corresponding author: Jiaquan Zhan

Received  June 2017 Revised  December 2017 Published  November 2018

In real game problems not all players can cooperate directly, games with communication structures introduced by Myerson in 1977 can deal with these problems quite well. More recently, this concept has been introduced into fuzzy games. In this paper, games on (fuzzy) communication structures were studied. We proved that if a coalitional game has a nonempty core, then the game restricted on an n-person connected graph also has a nonempty core. Further, the fuzzy game restricted on the n-person connected graph also has a nonempty core. Moreover, we proved the above two cores are identical and the core of the coalitional game is included in them. In addition, optimal fuzzy communication structures of fuzzy games were studied. We showed that the optimal communication structures do exist and proposed three allocating methods. In the end, a full illustrating example was given.

Keywords: Fuzzy games, fuzzy communication structures, core, myerson value.
Mathematics Subject Classification: 91A12, 91A43.
Citation: Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
References:
[1]

J. P. Aubin, Cooperative fuzzy games, Mathematics of Operations Research, 6 (1981), 1-13.  doi: 10.1287/moor.6.1.1.  Google Scholar

[2]

Y. Chen, Mean square exponential stability of uncertain singular stochastic systems with discrete and distributed delays, Journal of Interdisciplinary Mathematics, 20, 13-26. Google Scholar

[3]

D. B. Gillies, Solutions to general non-zero-sum games, Contributions to the Theory of Games IV, 4 (1959), 47-85.   Google Scholar

[4]

M. Grabisch, Games induced by the partitioning of a graph, Annals of Operations Research, 201 (2012), 229-249.  doi: 10.1007/s10479-012-1200-8.  Google Scholar

[5]

S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054.  Google Scholar

[6]

A. Jiménez-Losada, J. R. Fernández and M. Ordóñez, Myerson values for games with fuzzy communication structure, Fuzzy sets and systems, 213 (2013), 74-90.  doi: 10.1016/j.fss.2012.05.013.  Google Scholar

[7]

A. Jiménez-Losada, J. R. Fernández, M. Ordóñez and M. Grabisch, Games on fuzzy communication structures with choquet players, European Journal of Operational Research, 207 (2010), 836-847.  doi: 10.1016/j.ejor.2010.06.014.  Google Scholar

[8]

Y. H. Kou, Study on the property developers dynamic capabilities from the perspective of structural innovation, Journal of Discrete Mathematical Sciences & Cryptography, 19 (2016), 591-606.   Google Scholar

[9]

V. G. Luneeva Olga L.—Zakirova, Integration of mathematical and natural-science knowledge in school students' project-based activity., Eurasia Journal of Mathematics Science & Technology Education, 13 (2017), 2821-2840. Google Scholar

[10]

I. Mikhailova, A proof of zhil'tsov's theorem on decidability of equational theory of epigroups, Discrete Mathematics and Theoretical Computer Science, 17 (2016), 179-201.   Google Scholar

[11]

N. Ngoc-Thanh, M. Nunez and B. Trawinski, Collective intelligent information and database systems, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 32. Google Scholar

[12]

B. Peleg and P. Sudhölter, Introduction to the theory of cooperative games, Mathematical Programming and Operations Research, 34. Springer, Berlin, 2007.  Google Scholar

[13]

L. S. Shapley, On balanced sets and cores, 14, 453-460. Google Scholar

[14]

L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317.   Google Scholar

[15]

M. Tsurumi, T. Tanino and M. Inuiguchi, A shapley function on a class of cooperative fuzzy games, European Journal of Operational Research, 129 (2001), 596-618.  doi: 10.1016/S0377-2217(99)00471-3.  Google Scholar

[16]

J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947.  Google Scholar

[17]

J. Zhan and Z. Qiang, Optimal fuzzy coalition structure and solution concepts of a class of fuzzy games, in International Conference on Computer Science and Service System, 2011, 3971-3975. Google Scholar

show all references

References:
[1]

J. P. Aubin, Cooperative fuzzy games, Mathematics of Operations Research, 6 (1981), 1-13.  doi: 10.1287/moor.6.1.1.  Google Scholar

[2]

Y. Chen, Mean square exponential stability of uncertain singular stochastic systems with discrete and distributed delays, Journal of Interdisciplinary Mathematics, 20, 13-26. Google Scholar

[3]

D. B. Gillies, Solutions to general non-zero-sum games, Contributions to the Theory of Games IV, 4 (1959), 47-85.   Google Scholar

[4]

M. Grabisch, Games induced by the partitioning of a graph, Annals of Operations Research, 201 (2012), 229-249.  doi: 10.1007/s10479-012-1200-8.  Google Scholar

[5]

S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054.  Google Scholar

[6]

A. Jiménez-Losada, J. R. Fernández and M. Ordóñez, Myerson values for games with fuzzy communication structure, Fuzzy sets and systems, 213 (2013), 74-90.  doi: 10.1016/j.fss.2012.05.013.  Google Scholar

[7]

A. Jiménez-Losada, J. R. Fernández, M. Ordóñez and M. Grabisch, Games on fuzzy communication structures with choquet players, European Journal of Operational Research, 207 (2010), 836-847.  doi: 10.1016/j.ejor.2010.06.014.  Google Scholar

[8]

Y. H. Kou, Study on the property developers dynamic capabilities from the perspective of structural innovation, Journal of Discrete Mathematical Sciences & Cryptography, 19 (2016), 591-606.   Google Scholar

[9]

V. G. Luneeva Olga L.—Zakirova, Integration of mathematical and natural-science knowledge in school students' project-based activity., Eurasia Journal of Mathematics Science & Technology Education, 13 (2017), 2821-2840. Google Scholar

[10]

I. Mikhailova, A proof of zhil'tsov's theorem on decidability of equational theory of epigroups, Discrete Mathematics and Theoretical Computer Science, 17 (2016), 179-201.   Google Scholar

[11]

N. Ngoc-Thanh, M. Nunez and B. Trawinski, Collective intelligent information and database systems, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, 32. Google Scholar

[12]

B. Peleg and P. Sudhölter, Introduction to the theory of cooperative games, Mathematical Programming and Operations Research, 34. Springer, Berlin, 2007.  Google Scholar

[13]

L. S. Shapley, On balanced sets and cores, 14, 453-460. Google Scholar

[14]

L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317.   Google Scholar

[15]

M. Tsurumi, T. Tanino and M. Inuiguchi, A shapley function on a class of cooperative fuzzy games, European Journal of Operational Research, 129 (2001), 596-618.  doi: 10.1016/S0377-2217(99)00471-3.  Google Scholar

[16]

J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947.  Google Scholar

[17]

J. Zhan and Z. Qiang, Optimal fuzzy coalition structure and solution concepts of a class of fuzzy games, in International Conference on Computer Science and Service System, 2011, 3971-3975. Google Scholar

Figure 1.  $\gamma$ and its partion by level
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