# American Institute of Mathematical Sciences

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

## Cores and optimal fuzzy communication structures of fuzzy games

 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.

Citation: Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete and 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. [2] Y. Chen, Mean square exponential stability of uncertain singular stochastic systems with discrete and distributed delays, Journal of Interdisciplinary Mathematics, 20, 13-26. [3] D. B. Gillies, Solutions to general non-zero-sum games, Contributions to the Theory of Games IV, 4 (1959), 47-85. [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. [5] S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054. [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. [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. [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. [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. [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. [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. [12] B. Peleg and P. Sudhölter, Introduction to the theory of cooperative games, Mathematical Programming and Operations Research, 34. Springer, Berlin, 2007. [13] L. S. Shapley, On balanced sets and cores, 14, 453-460. [14] L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. [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. [16] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947. [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.

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. [2] Y. Chen, Mean square exponential stability of uncertain singular stochastic systems with discrete and distributed delays, Journal of Interdisciplinary Mathematics, 20, 13-26. [3] D. B. Gillies, Solutions to general non-zero-sum games, Contributions to the Theory of Games IV, 4 (1959), 47-85. [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. [5] S. Hart and A. Mas-Colell, Potential, value, and consistency, Econometrica, 57 (1989), 589-614.  doi: 10.2307/1911054. [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. [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. [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. [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. [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. [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. [12] B. Peleg and P. Sudhölter, Introduction to the theory of cooperative games, Mathematical Programming and Operations Research, 34. Springer, Berlin, 2007. [13] L. S. Shapley, On balanced sets and cores, 14, 453-460. [14] L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. [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. [16] J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947. [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.
$\gamma$ and its partion by level
 [1] İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics and Games, 2021, 8 (3) : 267-275. doi: 10.3934/jdg.2021010 [2] Linh Nguyen, Irina Perfilieva, Michal Holčapek. Boundary value problem: Weak solutions induced by fuzzy partitions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 715-732. doi: 10.3934/dcdsb.2019263 [3] Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial and Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581 [4] Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699 [5] Natalia Skripnik. Averaging of fuzzy integral equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118 [6] Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619 [7] Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008 [8] Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099 [9] Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117 [10] Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611 [11] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [12] George A. Anastassiou. Fractional Ostrowski-Sugeno Fuzzy univariate inequalities. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3305-3317. doi: 10.3934/dcdss.2020111 [13] Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial and Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158 [14] Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial and Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157 [15] Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007 [16] Jian Luo, Xueqi Yang, Ye Tian, Wenwen Yu. Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2743-2756. doi: 10.3934/jimo.2019078 [17] Seiyed Hadi Abtahi, Hamidreza Rahimi, Maryam Mosleh. Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network. Mathematical Foundations of Computing, 2021, 4 (3) : 209-219. doi: 10.3934/mfc.2021013 [18] Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457 [19] Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial and Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046 [20] Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2018  doi: 10.3934/bdia.2017018

2020 Impact Factor: 2.425

## Tools

Article outline

Figures and Tables