Core of cooperative fuzzy games with coalition structures

Fengye Wang; Yuanyuan Huang; Youlin Shang; Zhihua Zheng; Fengye Wang; Yuanyuan Huang; Youlin Shang; Zhihua Zheng

doi:10.3934/math.20251174

AIMS Mathematics

2025, Volume 10, Issue 11: 26697-26716. doi: 10.3934/math.20251174

Previous Article Next Article

Research article Topical Sections

Core of cooperative fuzzy games with coalition structures

1.
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
2.
School of Logistics and Management Engineering, Yunnan University of Finance and Economics, Yunnan, 650221, China

Received: 16 July 2025 Revised: 01 October 2025 Accepted: 28 October 2025 Published: 18 November 2025
MSC : 91A12, 91A35

This paper develops an extended model of cooperative fuzzy games with coalition structures, where fuzzy coalitions are characterized by real-valued functions. Moving beyond existing studies on crisp coalition structures, our framework integrates fuzzy set principles with coalitional cooperation to model partial participation in structured games. Our contributions are fourfold: (1) a formal core definition for such games, extending the Aubin core; (2) construction of the superadditive cover and establishment of core non-emptiness conditions; (3) an axiomatic characterization showing the core is the unique solution that satisfies non-emptiness, individual rationality, weak reduced game property, and superadditivity; and (4) a domination-core concept proven equivalent to the conventional core under specific conditions. A concise numerical example validates these theoretical findings, illustrating core allocation, non-emptiness, and the core-cover equivalence in scenarios of partial cooperation. These results unify fuzzy games with structured coalitions, offering new insights into the stability and allocation in games with graded participation and advancing the theory of cooperative decision-making under uncertainty.
- cooperative fuzzy game,
- coalition structure,
- core,
- domination-core
Citation: Fengye Wang, Yuanyuan Huang, Youlin Shang, Zhihua Zheng. Core of cooperative fuzzy games with coalition structures[J]. AIMS Mathematics, 2025, 10(11): 26697-26716. doi: 10.3934/math.20251174

Related Papers:

Abstract

This paper develops an extended model of cooperative fuzzy games with coalition structures, where fuzzy coalitions are characterized by real-valued functions. Moving beyond existing studies on crisp coalition structures, our framework integrates fuzzy set principles with coalitional cooperation to model partial participation in structured games. Our contributions are fourfold: (1) a formal core definition for such games, extending the Aubin core; (2) construction of the superadditive cover and establishment of core non-emptiness conditions; (3) an axiomatic characterization showing the core is the unique solution that satisfies non-emptiness, individual rationality, weak reduced game property, and superadditivity; and (4) a domination-core concept proven equivalent to the conventional core under specific conditions. A concise numerical example validates these theoretical findings, illustrating core allocation, non-emptiness, and the core-cover equivalence in scenarios of partial cooperation. These results unify fuzzy games with structured coalitions, offering new insights into the stability and allocation in games with graded participation and advancing the theory of cooperative decision-making under uncertainty.

References

[1]	B. Peleg, An axiomatization of the core of cooperative games without side payments, J. Math. Econ., 14 (1985), 203–214. http://doi.org/10.1016/0304-4068(85)90020-5 doi: 10.1016/0304-4068(85)90020-5
[2]	K. Tadenuma, Reduced games, consistency, and the core, Int. J. Game Theory, 20 (1992), 325–334. http://doi.org/10.1007/BF01271129
[3]	R. Serrano, O. Volij, Axiomatizations of neoclassical concepts for economies, J. Math. Econ., 30 (1998), 87–108. http://doi.org/10.1016/S0304-4068(97)00025-6 doi: 10.1016/S0304-4068(97)00025-6
[4]	Y. A. Hwang, P. Sudhölter, Axiomatizations of the core on the universal domain and other natural domains, Int. J. Game Theory, 29 (2001), 597–623. http://doi.org/10.1007/s001820100060 doi: 10.1007/s001820100060
[5]	J. P. Aubin, Cooperative fuzzy games, Math. Oper. Res., 6 (1981), 1–13. https://doi.org/10.1287/moor.6.1.1
[6]	J. P. Aubin, Mathematical methods of game and economic theory, 2 Eds., Amsterdam: North-Holland, 1982.
[7]	M. Tsurumi, T. Tanino, M. Inuiguchi, The core and the related solution concepts in cooperative fuzzy games, Journal of Japan Society for Fuzzy Theory and Systems, 12 (2000), 193–202. https://doi.org/10.3156/jfuzzy.12.1_193 doi: 10.3156/jfuzzy.12.1_193
[8]	X. H. Yu, Q. Zhang, The fuzzy core in games with fuzzy coalitions, J. Comput. Appl. Math., 230 (2009), 173–186. http://doi.org/10.1016/j.cam.2008.11.004 doi: 10.1016/j.cam.2008.11.004
[9]	Y. A. Hwang, Fuzzy games: A characterization of the core, Fuzzy Set. Syst., 158 (2007), 2480–2493. http://doi.org/10.1016/j.fss.2007.03.009 doi: 10.1016/j.fss.2007.03.009
[10]	S. Muto, S. Ishihara, E. Fukuda, S. H. Tijs, R. Branzei, Generalized cores and stable sets for fuzzy games, Int. Game Theory Rev., 8 (2006), 95–109. http://doi.org/10.1142/s0219198906000801 doi: 10.1142/s0219198906000801
[11]	V. A. Vasil'Ev, On the core of the Aubin extension of an almost positive game, Sib. Adv. Math., 35 (2025), 79–86. http://doi.org/10.1134/S1055134425010067 doi: 10.1134/S1055134425010067
[12]	X. H. Yu, Fuzzy cooperative game with intersecting priori coalition: A generalized configuration value, Fuzzy Set. Syst., 471 (2023), 108684. http://doi.org/10.1016/j.fss.2023.108684 doi: 10.1016/j.fss.2023.108684
[13]	V. B. Vilkov, A. I. Dergachev, A. K. Chernykh, M. S. Abu-Khasan, On the concept of solving a fuzzy cooperative game with side payments, 2020 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon), Vladivostok, Russia, 2020, 1–8. http://doi.org/10.1109/FarEastCon50210.2020.9271558
[14]	R. J. Aumann, J. H. Drèze, Cooperative games with coalition structures, Int. J. Game Theory, 3 (1974), 217–237. http://doi.org/10.1007/BF01766876 doi: 10.1007/BF01766876
[15]	G. Owen, Values of games with a priori unions, In: Mathematical economics and game theory, Heidelberg: Springer, 1977, 76–88. https://doi.org/10.1007/978-3-642-45494-3_7
[16]	G. Owen, Characterization of the Banzhaf-Coleman index, SIAM J. Appl. Math., 35 (1978), 315–327. http://doi.org/10.1137/0135026 doi: 10.1137/0135026
[17]	X. H. Yu, M. K. He, H. X. Sun, Z. Zhou, Uncertain coalition structure game with payoff of belief structure, Appl. Math. Comput., 372 (2020), 125000. https://doi.org/10.1016/j.amc.2019.125000 doi: 10.1016/j.amc.2019.125000
[18]	H. X. Sun, Q. Zhang, An axiomatization of probabilistic Owen value for games with coalition structure, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 571–582. http://doi.org/10.1007/s10255-014-0403-y doi: 10.1007/s10255-014-0403-y
[19]	X. H. Yu, Extension of owen value for the game with a coalition structure under the limited feasible coalition, Soft Comput., 25 (2021), 6139–6156. http://doi.org/10.1007/s00500-021-05604-9 doi: 10.1007/s00500-021-05604-9
[20]	F. Y. Meng, Q. Zhang, Y. Wang, Cooperative fuzzy games with a coalition structure and interval payoffs, Int. J. Comput. Intell. Syst., 6 (2013), 548–558. http://doi.org/10.1080/18756891.2013.795395 doi: 10.1080/18756891.2013.795395
[21]	M. Maschler, B. Peleg, The structure of the kernel of a cooperative game, SIAM J. Appl. Math., 15 (1967), 569–604. https://doi.org/10.1137/0115050 doi: 10.1137/0115050
[22]	B. Peleg, P. Sudhölter, Introduction to the theory of cooperative games, 2 Eds., Berlin Heidelberg: Springer, 2007. http://doi.org/10.1007/978-3-540-72945-7
[23]	M. Akram, F. Ilyas, M. Deveci, Interval rough integrated SWARA-ELECTRE model: An application to machine tool remanufacturing, Expert Syst. Appl., 238 (2024), 122067. https://doi.org/10.1016/j.eswa.2023.122067 doi: 10.1016/j.eswa.2023.122067
[24]	L. Y. Zhou, S. Abdullah, H. Zafar, S. Muhammad, A. Qadir, H. S. Huang, Analysis of artificial neural network based on pq-rung Orthopair fuzzy linguistic muirhead mean operators, Expert Syst. Appl., 276 (2025), 127157. https://doi.org/10.1016/j.eswa.2025.127157 doi: 10.1016/j.eswa.2025.127157
[25]	Z. F. Li, W. Huang, X. C. Gong, X. Y. Luo, K. Xiao, H. L. Deng, et al., Decoupled semantic graph neural network for knowledge graph embedding, Neurocomputing, 611 (2025), 128614. https://doi.org/10.1016/j.neucom.2024.128614 doi: 10.1016/j.neucom.2024.128614
[26]	J. Sun, G. P. Wang, J. Yun, L. Liu, Q. H. Shan, J. X. Zhang, Cooperative output regulation of multi-agent systems via energy-dependent intermittent event-triggered compensator approach, J. Franklin I., 362 (2025), 107868. https://doi.org/10.1016/j.jfranklin.2025.107868 doi: 10.1016/j.jfranklin.2025.107868

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)