{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T09:00:55Z","timestamp":1790586055859,"version":"4.1.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T00:00:00Z","timestamp":1766188800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T00:00:00Z","timestamp":1766188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Syst Sci Complex"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1007\/s11424-025-5077-4","type":"journal-article","created":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T04:18:59Z","timestamp":1766204339000},"page":"2188-2205","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Calculation and Convergence of Pareto-Nash Equilibria for Multicriteria Games"],"prefix":"10.1007","volume":"39","author":[{"given":"Shihua","family":"Fu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun-e","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ling","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xueying","family":"Nie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ya-nan","family":"Pan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoyu","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,12,20]]},"reference":[{"key":"5077_CR1","volume-title":"Theory of Games and Economic Behavior","author":"J Von Neumann","year":"1944","unstructured":"Von Neumann J and Morgenstern O, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944."},{"issue":"5659","key":"5077_CR2","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1126\/science.1093411","volume":"303","author":"A Nowak","year":"2004","unstructured":"Nowak A and Sigmund K, Evolutionary dynamics of biological games, Science, 2004, 303(5659): 793\u2013799.","journal-title":"Science"},{"issue":"6","key":"5077_CR3","doi-asserted-by":"publisher","first-page":"3444","DOI":"10.1109\/TCSS.2022.3209877","volume":"10","author":"C Q Qu","year":"2023","unstructured":"Qu C Q, Ji C L, and Zhang M, Evolutionary game model with group decision-making in signed social networks, IEEE Transactions on Computational Social Systems, 2023, 10(6): 3444\u20133453.","journal-title":"IEEE Transactions on Computational Social Systems"},{"key":"5077_CR4","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1109\/TASE.2023.3336640","volume":"22","author":"J H Wang","year":"2025","unstructured":"Wang J H, Liu M, and Wu H, The demand-side management and control of smart grids based on weighted network congestion games, IEEE Transactions on Automation Science and Engineering, 2025, 22): 43\u201352.","journal-title":"IEEE Transactions on Automation Science and Engineering"},{"issue":"1","key":"5077_CR5","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1016\/j.eswa.2011.07.017","volume":"39","author":"P Pendharkar","year":"2012","unstructured":"Pendharkar P, Game theoretical applications for multi-agent systems, Expert Systems with Applications, 2012, 39(1): 273\u2013279.","journal-title":"Expert Systems with Applications"},{"issue":"1","key":"5077_CR6","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1073\/pnas.36.1.48","volume":"36","author":"J F Nash","year":"1950","unstructured":"Nash J F, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 1950, 36(1): 48\u201349.","journal-title":"Proceedings of the National Academy of Sciences of the United States of America"},{"issue":"6","key":"5077_CR7","doi-asserted-by":"publisher","first-page":"2292","DOI":"10.1007\/s11424-023-2076-1","volume":"36","author":"Y T Zheng","year":"2023","unstructured":"Zheng Y T, Li C X, and Feng J E, Stability analysis of networked evolutionary games with profile-dependent delays, Journal of Systems Science & Complexity, 2023, 36(6): 2292\u20132308.","journal-title":"Journal of Systems Science & Complexity"},{"issue":"1","key":"5077_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2140\/pjm.1956.6.1","volume":"6","author":"D Blackwell","year":"1956","unstructured":"Blackwell D, An analog of the minimax theorem for vector payoffs, Pacific Journal of Mathematics, 1956, 6(1): 1\u20138.","journal-title":"Pacific Journal of Mathematics"},{"issue":"11","key":"5077_CR9","doi-asserted-by":"publisher","first-page":"1105","DOI":"10.1016\/j.aml.2007.12.009","volume":"21","author":"M Radjef","year":"2008","unstructured":"Radjef M and Fahem K, A note on ideal Nash equilibrium in multicriteria games, Applied Mathematics Letters, 2008, 21(11): 1105\u20131111.","journal-title":"Applied Mathematics Letters"},{"issue":"2","key":"5077_CR10","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/s10726-018-9608-4","volume":"28","author":"A Zapata","year":"2019","unstructured":"Zapata A, M\u00e1rmol A, Monroy L, et al., A maxmin approach for the equilibria of vector-valued games, Group Decision and Negotiation, 2019, 28(2): 415\u2013432.","journal-title":"Group Decision and Negotiation"},{"issue":"2","key":"5077_CR11","doi-asserted-by":"publisher","first-page":"1634","DOI":"10.1016\/j.jmaa.2017.01.002","volume":"457","author":"J Puerto","year":"2018","unstructured":"Puerto J and Perea F, On minimax and Pareto optimal security payoffs in multicriteria games, Journal of Mathematical Analysis and Applications, 2018, 457(2): 1634\u20131648.","journal-title":"Journal of Mathematical Analysis and Applications"},{"issue":"1","key":"5077_CR12","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1007\/s11750-007-0008-1","volume":"15","author":"F Patrone","year":"2007","unstructured":"Patrone F, Lucia P, and Tijs S, Multicriteria games and potentials, Top, 2007, 15(1): 138\u2013145.","journal-title":"Top"},{"issue":"6","key":"5077_CR13","doi-asserted-by":"publisher","first-page":"2131","DOI":"10.1007\/s11424-022-1064-1","volume":"35","author":"Y H Wang","year":"2022","unstructured":"Wang Y H and Li H T, Algebraic verification of finite group-based potential games with vector payoffs, Journal of Systems Science & Complexity, 2022, 35(6): 2131\u20132144.","journal-title":"Journal of Systems Science & Complexity"},{"issue":"5","key":"5077_CR14","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1134\/S0005117921050131","volume":"82","author":"A Rettieva","year":"2021","unstructured":"Rettieva A, Solutions of dynamic multicriteria games: Classical and untraditional approaches, Automation and Remote Control, 2021, 82(5): 902\u2013910.","journal-title":"Automation and Remote Control"},{"issue":"3","key":"5077_CR15","doi-asserted-by":"publisher","first-page":"1472","DOI":"10.1002\/asjc.3274","volume":"26","author":"L Yu","year":"2024","unstructured":"Yu L, Fu S H, Pan J, et al., Weighted Nash equilibrium of incomplete-profile networked evolutionary games with multiple payoffs, Asian Journal of Control, 2024, 26(3): 1472\u20131482.","journal-title":"Asian Journal of Control"},{"issue":"1","key":"5077_CR16","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1002\/nav.3800060107","volume":"6","author":"L S Shapley","year":"1959","unstructured":"Shapley L S and Rigby F D, Equilibrium points in games with vector payoffs, Naval Research Logistics Quarterly, 1959, 6(1): 57\u201361.","journal-title":"Naval Research Logistics Quarterly"},{"issue":"3","key":"5077_CR17","first-page":"115","volume":"49","author":"D Lozovanu","year":"2005","unstructured":"Lozovanu D, Solomon D, and Zelikovsky A, Multiobjective games and determining Pareto-Nash equilibria, Matematica, 2005, 49(3): 115\u2013122.","journal-title":"Matematica"},{"issue":"5","key":"5077_CR18","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1016\/S0893-9659(02)80006-4","volume":"15","author":"H Yang","year":"2002","unstructured":"Yang H and Yu J, Essential components of the set of weakly Pareto-Nash equilibrium points, Applied Mathematics Letters, 2002, 15(5): 553\u2013560.","journal-title":"Applied Mathematics Letters"},{"key":"5077_CR19","doi-asserted-by":"publisher","first-page":"1689732","DOI":"10.1155\/2022\/1689732","volume":"2022","author":"L P Liu","year":"2022","unstructured":"Liu L P, Jia W S, and Zhou L, Generic stability of the weakly Pareto-Nash equilibrium with strategy transformational barriers, Journal of Function Spaces, 2022, 2022): 1689732.","journal-title":"Journal of Function Spaces"},{"key":"5077_CR20","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1007\/s10479-017-2570-8","volume":"267","author":"N Yousfi-Halimi","year":"2018","unstructured":"Yousfi-Halimi N, Radjef M S, and Slimani H, Refinement of pure Pareto Nash equilibria in finite multicriteria games using preference relations, Annals of Operations Research, 2018, 267): 607\u2013628.","journal-title":"Annals of Operations Research"},{"key":"5077_CR21","doi-asserted-by":"publisher","first-page":"4972","DOI":"10.1016\/j.jfranklin.2021.04.023","volume":"358","author":"Y Q Hao","year":"2021","unstructured":"Hao Y Q and Cheng D Z, Optimization of multi-criteria facility-based systems via vector potential approach, Journal of the Franklin Institute, 2021, 358): 4972\u20134993.","journal-title":"Journal of the Franklin Institute"},{"key":"5077_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-85729-097-7","volume-title":"Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach","author":"D Z Cheng","year":"2011","unstructured":"Cheng D Z, Qi H S, and Li Z Q, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach, Springer, London, 2011."},{"key":"5077_CR23","doi-asserted-by":"publisher","first-page":"126380","DOI":"10.1016\/j.amc.2021.126380","volume":"409","author":"P L Guo","year":"2021","unstructured":"Guo P L and Han C D, Nash equilibrium and group strategy consensus of networked evolutionary game with coupled social groups, Applied Mathematics and Computation, 2021, 409): 126380.","journal-title":"Applied Mathematics and Computation"},{"issue":"4","key":"5077_CR24","doi-asserted-by":"publisher","first-page":"818","DOI":"10.1109\/JAS.2016.7510259","volume":"5","author":"G D Zhao","year":"2018","unstructured":"Zhao G D, Wang Y Z, and Li H T, A matrix approach to the modeling and analysis of networked evolutionary games with time delays, IEEE\/CAA Journal of Automatica Sinica, 2018, 5(4): 818\u2013826.","journal-title":"IEEE\/CAA Journal of Automatica Sinica"},{"key":"5077_CR25","doi-asserted-by":"publisher","first-page":"101396","DOI":"10.1016\/j.nahs.2023.101396","volume":"50","author":"D Wang","year":"2023","unstructured":"Wang D, Fu S H, Wang J J, et al., The stability degree analysis of profiles for networked evolutionary games with switching topology, Nonlinear Analysis-Hybrid Systems, 2023, 50): 101396.","journal-title":"Nonlinear Analysis-Hybrid Systems"},{"issue":"6","key":"5077_CR26","doi-asserted-by":"publisher","first-page":"3975","DOI":"10.1109\/TAC.2023.3340558","volume":"69","author":"H T Li","year":"2024","unstructured":"Li H T, Wang S L, and Li W R, Aggregation method to strategy consensus of large-size networked evolutionary matrix games, IEEE Transactions on Automatic Control, 2024, 69(6): 3975\u20133981.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"11","key":"5077_CR27","doi-asserted-by":"publisher","first-page":"7584","DOI":"10.1109\/TAC.2024.3391652","volume":"69","author":"C X Li","year":"2024","unstructured":"Li C X, Li A M, Wu Y H, et al., Logical dynamic games: Models, equilibria, and potentials, IEEE Transactions on Automatic Control, 2024, 69(11): 7584\u20137599.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"2","key":"5077_CR28","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1109\/TAC.2020.2985302","volume":"66","author":"Y Q Guo","year":"2021","unstructured":"Guo Y Q, Shen Y W, and Gui W H, Asymptotical stability of logic dynamical systems with random impulsive disturbances, IEEE Transactions on Automatic Control, 2021, 66(2): 513\u2013525.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"3","key":"5077_CR29","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1007\/s11424-023-1162-8","volume":"36","author":"C F Jiang","year":"2023","unstructured":"Jiang C F, Wang B, Fu S H, et al., Observability of periodically switched Boolean control networks, Journal of Systems Science & Complexity, 2023, 36(3): 985\u20131001.","journal-title":"Journal of Systems Science & Complexity"},{"issue":"4","key":"5077_CR30","doi-asserted-by":"publisher","first-page":"2206","DOI":"10.1109\/TCYB.2022.3219522","volume":"54","author":"Y T Chen","year":"2024","unstructured":"Chen Y T, Yang J Q, Li Z Q, et al., State estimation via designing controller and state estimation-based stabilization for Boolean control networks, IEEE Transactions on Cybernetics, 2024, 54(4): 2206\u20132215.","journal-title":"IEEE Transactions on Cybernetics"},{"issue":"2","key":"5077_CR31","first-page":"159","volume":"38","author":"X J Cao","year":"2025","unstructured":"Cao X J, Fu S H, Zhao J L, et al., The solvability of output regulation problem for Boolean networks with disturbance inputs under one-bit function perturbation, Journal of Liaocheng University (Natural Science Edition), 2025, 38(2): 159\u2013169.","journal-title":"Journal of Liaocheng University (Natural Science Edition)"},{"issue":"6","key":"5077_CR32","doi-asserted-by":"publisher","first-page":"6450","DOI":"10.1002\/mma.8916","volume":"46","author":"X L Fan","year":"2023","unstructured":"Fan X L, Li Y, Ding W X, et al., Semi-tensor product of quaternion matrices and its application, Mathematical Methods in the Applied Sciences, 2023, 46(6): 6450\u20136462.","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"7","key":"5077_CR33","doi-asserted-by":"publisher","first-page":"6036","DOI":"10.1002\/mma.9905","volume":"47","author":"W X Ding","year":"2024","unstructured":"Ding W X, Li Y, Liu Z H, et al., Algebraic method for LU decomposition in commutative quaternion based on semi-tensor product of matrices and application to strict image authentication, Mathematical Methods in the Applied Sciences, 2024, 47(7): 6036\u20136050.","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"5077_CR34","doi-asserted-by":"publisher","first-page":"116072","DOI":"10.1016\/j.cam.2024.116072","volume":"451","author":"A L Wei","year":"2024","unstructured":"Wei A L, Li Y, Ding W X, et al., A new structure-preserving algorithm based on the semi-tensor product of matrices for split quaternion matrix LDU decomposition and its applications, Journal of Computational and Applied Mathematics, 2024, 451): 116072.","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"9","key":"5077_CR35","doi-asserted-by":"publisher","first-page":"192203","DOI":"10.1007\/s11432-023-4044-8","volume":"67","author":"H T Li","year":"2024","unstructured":"Li H T, Liu Z Q, and Li W R, Nonsingularity of Grain-like cascade feedback shift registers subject to fault attacks, Science China Information Sciences, 2024, 67(9): 192203.","journal-title":"Science China Information Sciences"},{"key":"5077_CR36","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.automatica.2018.07.011","volume":"96","author":"J Q Lu","year":"2018","unstructured":"Lu J Q, Li M L, Huang T, et al., The transformation between the Galois NLFSRs and the Fibonacci NLFSRs via semi-tensor product of matrices, Automatica, 2018, 96): 393\u2013397.","journal-title":"Automatica"}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-025-5077-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11424-025-5077-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-025-5077-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T08:52:53Z","timestamp":1790585573000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11424-025-5077-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,20]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["5077"],"URL":"https:\/\/doi.org\/10.1007\/s11424-025-5077-4","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,20]]},"assertion":[{"value":"3 March 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 August 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 December 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}