{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T09:52:07Z","timestamp":1775037127003,"version":"3.50.1"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,12,16]],"date-time":"2025-12-16T00:00:00Z","timestamp":1765843200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,16]],"date-time":"2025-12-16T00:00:00Z","timestamp":1765843200000},"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 Optim Theory Appl"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10957-025-02902-2","type":"journal-article","created":{"date-parts":[[2025,12,16]],"date-time":"2025-12-16T12:52:59Z","timestamp":1765889579000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Further on Potential Differential Games and Its Applications in Multi-leader Multi-follower Differential Games"],"prefix":"10.1007","volume":"208","author":[{"given":"Asmamaw Msgan","family":"Ayele","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Addis Belete","family":"Zewde","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5494-040X","authenticated-orcid":false,"given":"Semu Mitiku","family":"Kassa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,16]]},"reference":[{"issue":"1","key":"2902_CR1","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1016\/S1474-6670(17)68353-8","volume":"5","author":"CI Chen","year":"1972","unstructured":"Chen, C.I., Cruz, J.B., Jr.: Differential games with different intervals of play. IFAC Proc. 5(1), 348\u2013354 (1972)","journal-title":"IFAC Proc."},{"key":"2902_CR2","doi-asserted-by":"publisher","first-page":"954","DOI":"10.1109\/TIFS.2022.3155294","volume":"17","author":"Z Cheng","year":"2022","unstructured":"Cheng, Z., Chen, G., Hong, Y.: Single-leader-multiple-followers stackelberg security game with hypergame framework. IEEE Trans. Inf. Forensics Secur. 17, 954\u2013969 (2022)","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"2902_CR3","volume-title":"Funct. Anal.","author":"F Clarke","year":"2013","unstructured":"Clarke, F.: Funct. Anal. Calculus of Variations and Optimal Control. Graduate Texts in Mathematics. Springer-Verlag, London (2013)"},{"key":"2902_CR4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511805127","volume-title":"Differential Games in Economics and Management Science","author":"E Dockner","year":"2000","unstructured":"Dockner, E.: Differential Games in Economics and Management Science. Cambridge University Press, United Kingdom (2000)"},{"issue":"2","key":"2902_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3182\/20090506-3-SF-4003.00002","volume":"42","author":"D Dragone","year":"2009","unstructured":"Dragone, D., Lambertini, L., Leitmann, G., Palestini, A.: Hamiltonian potential functions for differential games. IFAC Proc. Vol. 42(2), 1\u20138 (2009)","journal-title":"IFAC Proc. Vol."},{"key":"2902_CR6","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.automatica.2015.09.036","volume":"62","author":"D Dragone","year":"2015","unstructured":"Dragone, D., Lambertini, L., Leitmann, G., Palestini, A.: Hamiltonian potential functions for differential games. Automatica 62, 134\u2013138 (2015)","journal-title":"Automatica"},{"issue":"11","key":"2902_CR7","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1080\/02331934.2010.541457","volume":"61","author":"D Dragone","year":"2012","unstructured":"Dragone, D., Lambertini, L., Palestini, A.: Static and dynamic best-response potential functions for the non-linear cournot game. Optimization 61(11), 1283\u20131293 (2012)","journal-title":"Optimization"},{"key":"2902_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-1031-4","volume-title":"Dynamic Models of Advertising Competition","author":"G Erickson","year":"2003","unstructured":"Erickson, G.: Dynamic Models of Advertising Competition. International Series in Quantitative Marketing. Springer, New York, NY (2003)"},{"key":"2902_CR9","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF00939082","volume":"55","author":"C Fershtman","year":"1987","unstructured":"Fershtman, C.: Identification of classes of differential games for which the open loop is a degenerate feedback Nash equilibrium. J. Optim. Theory Appl. 55, 217\u2013231 (1987)","journal-title":"J. Optim. Theory Appl."},{"key":"2902_CR10","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1007\/s13235-017-0218-6","volume":"8","author":"A Fonseca-Morales","year":"2018","unstructured":"Fonseca-Morales, A., Hern\u00e1ndez-Lerma, O.: Potential differential games. Dyn. Games Appl. 8, 254\u2013279 (2018)","journal-title":"Dyn. Games Appl."},{"key":"2902_CR11","doi-asserted-by":"publisher","first-page":"2075","DOI":"10.1007\/s11425-016-0264-6","volume":"59","author":"D Gonz\u00e1lez-S\u00e1nchez","year":"2016","unstructured":"Gonz\u00e1lez-S\u00e1nchez, D., Hern\u00e1ndez-Lerma, O.: A survey of static and dynamic potential games. Sci. China Math. 59, 2075\u20132102 (2016)","journal-title":"Sci. China Math."},{"issue":"6","key":"2902_CR12","doi-asserted-by":"publisher","first-page":"2058","DOI":"10.1109\/TMC.2020.3035990","volume":"21","author":"S Jiang","year":"2020","unstructured":"Jiang, S., Li, X., Wu, J.: Multi-leader multi-follower stackelberg game in mobile blockchain mining. IEEE Trans. Mob. Comput. 21(6), 2058\u20132071 (2020)","journal-title":"IEEE Trans. Mob. Comput."},{"key":"2902_CR13","doi-asserted-by":"crossref","unstructured":"Jim\u00e9nez-Liz\u00e1rraga, M., Poznyak, A., Alcorta, M.A.: Leader-follower strategies for a multi-plant differential game. In 2008 American Control Conference, IEEE pp. 3839 \u2013 3844 (2008)","DOI":"10.1109\/ACC.2008.4587092"},{"key":"2902_CR14","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s10898-017-0502-4","volume":"68","author":"AM Kassa","year":"2017","unstructured":"Kassa, A.M., Kassa, S.M.: Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers. J. Global Optim. 68, 729\u2013747 (2017)","journal-title":"J. Global Optim."},{"key":"2902_CR15","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/978-3-030-52119-6_16","volume-title":"Bilevel Optimization: Advances and Next Challenges","author":"P Mehlitz","year":"2020","unstructured":"Mehlitz, P., Wachsmuth, G.: Bilevel optimal control: existence results and stationarity conditions. In: Dempe, S., Zemkoho, A. (eds.) Bilevel Optimization: Advances and Next Challenges, pp. 451\u2013484. Springer International Publishing, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-52119-6_16"},{"issue":"1","key":"2902_CR16","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1006\/game.1996.0044","volume":"14","author":"D Monderer","year":"1996","unstructured":"Monderer, D., Shapley, L.S.: Potential games. Games Econom. Behav. 14(1), 124\u2013143 (1996)","journal-title":"Games Econom. Behav."},{"issue":"2","key":"2902_CR17","doi-asserted-by":"publisher","first-page":"539","DOI":"10.4310\/PAMQ.2007.v3.n2.a7","volume":"3","author":"L Mou","year":"2007","unstructured":"Mou, L., Yong, J.: A variational formula for stochastic controls and some applications. Pure Appl. Math. Quart. 3(2), 539\u2013567 (2007)","journal-title":"Pure Appl. Math. Quart."},{"issue":"2","key":"2902_CR18","doi-asserted-by":"publisher","first-page":"522","DOI":"10.1016\/0022-247X(91)90348-4","volume":"161","author":"L Pan","year":"1991","unstructured":"Pan, L., Yong, J.: A differential game with multi-level of hierarchy. J. Math. Anal. Appl. 161(2), 522\u2013544 (1991)","journal-title":"J. Math. Anal. Appl."},{"key":"2902_CR19","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01737559","volume":"2","author":"RW Rosenthal","year":"1973","unstructured":"Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. Internat. J. Game Theory 2, 65\u201367 (1973)","journal-title":"Internat. J. Game Theory"},{"issue":"5","key":"2902_CR20","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1007\/BF00935665","volume":"11","author":"M Simman","year":"1973","unstructured":"Simman, M., Cruz, J.B., Jr.: On the stackerberg strategy in nonzero games. J. Optim. Theory Appl. 11(5), 533\u2013555 (1973)","journal-title":"J. Optim. Theory Appl."},{"key":"2902_CR21","doi-asserted-by":"publisher","DOI":"10.2307\/20075757","author":"ME Slade","year":"1989","unstructured":"Slade, M.E.: The fictitious payoff-function: two applications to dynamic games. Annales d\u2019\u00c9conomie et de Statistique (15\/16) (1989). https:\/\/doi.org\/10.2307\/20075757","journal-title":"Annales d\u2019\u00c9conomie et de Statistique (15\/16)"},{"key":"2902_CR22","volume-title":"What Does an Oligopoly Maximize?","author":"ME Slade","year":"1989","unstructured":"Slade, M.E.: What Does an Oligopoly Maximize? Universite Aix-Marseille III, Tech. rep. (1989)"},{"issue":"1","key":"2902_CR23","doi-asserted-by":"publisher","first-page":"45","DOI":"10.2307\/2950588","volume":"42","author":"ME Slade","year":"1994","unstructured":"Slade, M.E.: What does an oligopoly maximize? J. Ind. Econ. 42(1), 45\u201361 (1994)","journal-title":"J. Ind. Econ."},{"key":"2902_CR24","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1051\/proc\/202171155","volume":"71","author":"S Steffensen","year":"2021","unstructured":"Steffensen, S.: Multi-leader-follower potential games. ESAIM: Proc. Surv. 71, 155\u2013162 (2021)","journal-title":"ESAIM: Proc. Surv."},{"key":"2902_CR25","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s10957-015-0723-y","volume":"167","author":"N Tauchnitz","year":"2015","unstructured":"Tauchnitz, N.: The pontryagin maximum principle for nonlinear optimal control problems with infinite horizon. J. Optim. Theory Appl. 167, 27\u201348 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2902_CR26","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1002\/acs.2831","volume":"33","author":"KG Vamvoudakis","year":"2019","unstructured":"Vamvoudakis, K.G., Lewis, F.L., Dixon, W.E.: Open-loop stackelberg learning solution for hierarchical control problems. Int. J. Adapt. Control Signal Process. 33(2), 285\u2013299 (2019)","journal-title":"Int. J. Adapt. Control Signal Process."},{"issue":"6","key":"2902_CR27","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s40314-024-02861-w","volume":"43","author":"B Varga","year":"2024","unstructured":"Varga, B., Huang, D., Hohmann, S.: Identification methods for ordinal potential differential games. Comput. Appl. Math. 43(6), 343 (2024)","journal-title":"Comput. Appl. Math."},{"key":"2902_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijepes.2023.109577","volume":"155","author":"Y Wan","year":"2024","unstructured":"Wan, Y., Qin, J., Shi, Y., Fu, W., Xiao, F.: Stackelberg-Nash game approach for price-based demand response in retail electricity trading. Int. J. Electr. Power Energy Syst. 155, 109577 (2024)","journal-title":"Int. J. Electr. Power Energy Syst."},{"issue":"6","key":"2902_CR29","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1002\/net.3230230606","volume":"23","author":"BW Wie","year":"1993","unstructured":"Wie, B.W.: A differential game model of nash equilibrium on a congested traffic network. Networks 23(6), 557\u2013565 (1993)","journal-title":"Networks"},{"issue":"2","key":"2902_CR30","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1007\/s10957-022-02149-1","volume":"196","author":"SB Worku","year":"2023","unstructured":"Worku, S.B., Tsegaw, B.B., Kassa, S.M.: Multilevel reverse stackelberg differential games: existence and solution approach for affine strategies. J. Optim. Theory Appl. 196(2), 617\u2013640 (2023)","journal-title":"J. Optim. Theory Appl."},{"key":"2902_CR31","volume-title":"Cooperative Stochastic Differential Games","author":"DW Yeung","year":"2006","unstructured":"Yeung, D.W., Petrosyan, L.A.: Cooperative Stochastic Differential Games. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)"},{"key":"2902_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1466-3","volume-title":"Stochastic Controls: Hamiltonian Systems and HJB Equations","author":"J Yong","year":"1999","unstructured":"Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer Science & Business Media, Heidelberg (1999)"},{"issue":"5","key":"2902_CR33","doi-asserted-by":"publisher","first-page":"2915","DOI":"10.1051\/ro\/2021146","volume":"55","author":"AB Zewde","year":"2021","unstructured":"Zewde, A.B., Kassa, S.M.: Hierarchical multilevel optimization with multiple-leaders multiple-followers setting and nonseparable objectives. RAIRO-Op. Res. 55(5), 2915\u20132939 (2021)","journal-title":"RAIRO-Op. Res."},{"issue":"4","key":"2902_CR34","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1007\/s10287-021-00398-5","volume":"18","author":"AB Zewde","year":"2021","unstructured":"Zewde, A.B., Kassa, S.M.: Multilevel multi-leader multiple-follower games with nonseparable objectives and shared constraints. CMS 18(4), 455\u2013475 (2021)","journal-title":"CMS"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02902-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02902-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02902-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:23:59Z","timestamp":1775028239000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02902-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,16]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2902"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02902-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,16]]},"assertion":[{"value":"3 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 November 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 December 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors have declared to have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"62"}}