{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T15:04:00Z","timestamp":1779894240843,"version":"3.53.1"},"reference-count":32,"publisher":"Elsevier BV","issue":"9","license":[{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2027,5,4]],"date-time":"2027-05-04T00:00:00Z","timestamp":1809388800000},"content-version":"am","delay-in-days":337,"URL":"http:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of the Franklin Institute"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1016\/j.jfranklin.2026.108680","type":"journal-article","created":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T15:21:16Z","timestamp":1776525676000},"page":"108680","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linear-quadratic mean field games with multiple major players: A complexity analysis of heterogeneous and homogeneous architectures"],"prefix":"10.1016","volume":"363","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9459-4270","authenticated-orcid":false,"given":"Arman","family":"Sharifi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4130-7925","authenticated-orcid":false,"given":"Hao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"1","key":"10.1016\/j.jfranklin.2026.108680_bib0001","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s11537-007-0657-8","article-title":"Mean field games","volume":"2","author":"Lasry","year":"2007","journal-title":"Jpn. J. Math."},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108680_bib0002","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","article-title":"Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle","volume":"6","author":"Huang","year":"2006","journal-title":"Commun. Inf. Syst."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0003","series-title":"Mean Field Games and Mean Field Type Control Theory","volume":"101","author":"Bensoussan","year":"2013"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0004","series-title":"Probabilistic Theory of Mean Field Games with Applications I-II","author":"Carmona","year":"2018"},{"issue":"2","key":"10.1016\/j.jfranklin.2026.108680_bib0005","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1007\/s00199-020-01314-9","article-title":"Approximation and characterization of Nash equilibria of large games","volume":"73","author":"Carmona","year":"2022","journal-title":"Econ. Theory"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0006","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.automatica.2018.11.029","article-title":"An integral control formulation of mean field game based large scale coordination of loads in smart grids","volume":"100","author":"Kizilkale","year":"2019","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0007","article-title":"On a class of linear quadratic Gaussian quantilized mean field games","volume":"170","author":"Tchuendom","year":"2024","journal-title":"Automatica"},{"issue":"20","key":"10.1016\/j.jfranklin.2026.108680_bib0008","doi-asserted-by":"crossref","first-page":"5057","DOI":"10.3390\/en17205057","article-title":"Zero-trust zero-communication defence against hybrid cyberattacks in distributed energy resources using mean field reinforcement leaning","volume":"17","author":"Zhou","year":"2024","journal-title":"Energies"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0009","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2020.108835","article-title":"\u03f5-Nash mean-field games for general linear-quadratic systems with applications","volume":"114","author":"Xu","year":"2020","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2023.111420","article-title":"A mean field game approach for a class of linear quadratic discrete choice problems with congestion avoidance","volume":"160","author":"Toumi","year":"2024","journal-title":"Automatica"},{"issue":"6","key":"10.1016\/j.jfranklin.2026.108680_bib0011","doi-asserted-by":"crossref","first-page":"3225","DOI":"10.1137\/140985676","article-title":"Opinion dynamics in social networks through mean-field games","volume":"54","author":"Bauso","year":"2016","journal-title":"SIAM J. Contr. Optim."},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108680_bib0012","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.1109\/TAC.2007.904450","article-title":"Large-population cost-coupled LQG problems with nonuniform agents: individual-Mass behavior and decentralized \u03b5-Nash equilibria","volume":"52","author":"Huang","year":"2007","journal-title":"IEEE Trans. Autom. Contr."},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108680_bib0013","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1109\/TAC.2012.2215399","article-title":"Nash, social and centralized solutions to consensus problems via mean field control theory","volume":"58","author":"Nourian","year":"2012","journal-title":"IEEE Trans. Autom. Contr."},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108680_bib0014","doi-asserted-by":"crossref","first-page":"1736","DOI":"10.1109\/TAC.2012.2183439","article-title":"Social optima in mean field LQG control: centralized and decentralized strategies","volume":"57","author":"Huang","year":"2012","journal-title":"IEEE Trans. Autom. Contr."},{"issue":"12","key":"10.1016\/j.jfranklin.2026.108680_bib0015","doi-asserted-by":"crossref","first-page":"3784","DOI":"10.1109\/TAC.2016.2519501","article-title":"Backward mean-field linear-quadratic-Gaussian (LQG) games: full and partial information","volume":"61","author":"Huang","year":"2016","journal-title":"IEEE Trans. Autom. Contr."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0016","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2024.111518","article-title":"Discrete-time indefinite linear-quadratic mean field games and control: the finite-population case","volume":"162","author":"Liang","year":"2024","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0017","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2024.111821","article-title":"Linear\u2013quadratic mean-field game for stochastic systems with partial observation","volume":"171","author":"Li","year":"2025","journal-title":"Automatica"},{"issue":"5","key":"10.1016\/j.jfranklin.2026.108680_bib0018","doi-asserted-by":"crossref","first-page":"3318","DOI":"10.1137\/080735370","article-title":"Large-population LQG games involving a major player: the Nash certainty equivalence principle","volume":"48","author":"Huang","year":"2010","journal-title":"SIAM J. Contr. Optim."},{"issue":"2","key":"10.1016\/j.jfranklin.2026.108680_bib0019","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1080\/1350486X.2019.1603183","article-title":"Mean-field game strategies for optimal execution","volume":"26","author":"Huang","year":"2019","journal-title":"Appl. Math. Finance"},{"issue":"5","key":"10.1016\/j.jfranklin.2026.108680_bib0020","doi-asserted-by":"crossref","first-page":"5008","DOI":"10.1109\/TNSE.2024.3418832","article-title":"Hierarchical cooperation in LQ multi-population mean field game with its application to opinion evolution","volume":"11","author":"Ren","year":"2024","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"issue":"4","key":"10.1016\/j.jfranklin.2026.108680_bib0021","doi-asserted-by":"crossref","first-page":"3302","DOI":"10.1137\/120889496","article-title":"\u03f5-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents","volume":"51","author":"Nourian","year":"2013","journal-title":"SIAM J. Contr. Optim."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0022","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2022.110559","article-title":"LQG mean field games with a major agent: Nash certainty equivalence versus probabilistic approach","volume":"146","author":"Firoozi","year":"2022","journal-title":"Automatica"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0023","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.automatica.2018.08.008","article-title":"Linear quadratic mean field Stackelberg differential games","volume":"97","author":"Moon","year":"2018","journal-title":"Automatica"},{"issue":"8","key":"10.1016\/j.jfranklin.2026.108680_bib0024","doi-asserted-by":"crossref","first-page":"2241","DOI":"10.1109\/TAC.2014.2301576","article-title":"Hierarchical mean field games for multiagent systems with tracking-type costs: distributed \u03b5-Stackelberg equilibria","volume":"59","author":"Wang","year":"2014","journal-title":"IEEE Trans. Autom. Contr."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0025","series-title":"2024 American Control Conference (ACC)","first-page":"5075","article-title":"Multi-population mean field games with multiple major players: application to carbon emission regulations","author":"Dayanikli","year":"2024"},{"key":"10.1016\/j.jfranklin.2026.108680_bib0026","series-title":"Mean Field Models with Heterogeneous Agents","author":"Dayan\u0131kl\u0131","year":"2022"},{"issue":"1","key":"10.1016\/j.jfranklin.2026.108680_bib0027","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1109\/TCSS.2025.3588326","article-title":"Surrounding control based on optimal targets assignment and dynamic graph game","volume":"13","author":"Jia","year":"2025","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"issue":"4","key":"10.1016\/j.jfranklin.2026.108680_bib0028","doi-asserted-by":"crossref","DOI":"10.1007\/s11432-024-4295-8","article-title":"Manipulate holistic cooperation of population game: optimal induction strategy for a few irrational agents in social networks","volume":"68","author":"Wang","year":"2025","journal-title":"Sci. China Inf. Sci."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0029","first-page":"1","article-title":"Cooperative game-based intelligent actions making for constrained multi-agent system","author":"Yu","year":"2024","journal-title":"IEEE Trans. Artif. Intell."},{"issue":"4","key":"10.1016\/j.jfranklin.2026.108680_bib0030","doi-asserted-by":"crossref","first-page":"1397","DOI":"10.1109\/TAC.2019.2919111","article-title":"Linear quadratic mean field games: asymptotic solvability and relation to the fixed point approach","volume":"65","author":"Huang","year":"2019","journal-title":"IEEE Trans. Autom. Contr."},{"key":"10.1016\/j.jfranklin.2026.108680_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.automatica.2019.108774","article-title":"Linear quadratic mean field games with a major player: the multi-scale approach","volume":"113","author":"Ma","year":"2020","journal-title":"Automatica"},{"issue":"2","key":"10.1016\/j.jfranklin.2026.108680_bib0032","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1287\/moor.2018.0929","article-title":"Rare Nash equilibria and the price of anarchy in large static games","volume":"44","author":"Lacker","year":"2019","journal-title":"Math. Oper. Res."}],"container-title":["Journal of the Franklin Institute"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226002802?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226002802?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T14:18:55Z","timestamp":1779891535000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0016003226002802"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6]]},"references-count":32,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["S0016003226002802"],"URL":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108680","relation":{},"ISSN":["0016-0032"],"issn-type":[{"value":"0016-0032","type":"print"}],"subject":[],"published":{"date-parts":[[2026,6]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Linear-quadratic mean field games with multiple major players: A complexity analysis of heterogeneous and homogeneous architectures","name":"articletitle","label":"Article Title"},{"value":"Journal of the Franklin Institute","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108680","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Franklin Institute. Published by Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"108680"}}