{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T18:00:47Z","timestamp":1788285647554,"version":"build-2803163510"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We consider linear-quadratic-Gaussian (LQG) games with a major player and a large number of minor players. The major player has a significant influence on others. The minor players individually have negligible impact, but they collectively contribute mean field coupling terms in the individual dynamics and costs. To overcome the dimensionality difficulty and obtain decentralized strategies, the so-called Nash certainty equivalence methodology is applied. The control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving the mean field approximation. Subsequently, within the population limit the LQG game is decomposed into a family of limiting two-player games as each is locally seen by a representative minor player. Next, when solving these limiting two-player games, we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closed loop of a large number of minor players. This procedure leads to a set of decentralized strategies for the original LQG game, which is an $\\varepsilon$-Nash equilibrium.<\/jats:p>","DOI":"10.1137\/080735370","type":"journal-article","created":{"date-parts":[[2010,1,29]],"date-time":"2010-01-29T18:15:53Z","timestamp":1264788953000},"page":"3318-3353","source":"Crossref","is-referenced-by-count":276,"title":["Large-Population LQG Games Involving a Major Player: The Nash Certainty Equivalence Principle"],"prefix":"10.1137","volume":"48","author":[{"given":"Minyi","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,1,29]]},"reference":[{"key":"R1","unstructured":"R. J. Aumann,\n                      Game theory\n                      , in The New Palgrave: A Dictionary of Economics, J. Eatwell, M. Milgate, and P. Newman, eds., Vol. 2, Macmillan, London, 1987, pp. 460\u2013482."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1080\/00207178608933518"},{"key":"R3","unstructured":"T. Ba\u015far and G. J. Olsder,\n                      Dynamic Noncooperative Game Theory\n                      , 2nd ed., Academic Press, London, UK, 1995."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0327030"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0403823101"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.76.031127"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0312037"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"A. Bensoussan,\n                      Stochastic Control of Partially Observable Systems\n                      , Cambridge University Press, Cambridge, UK, 1992.","DOI":"10.1017\/CBO9780511526503"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/050639089"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0377-2217(94)00232-2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1287\/moor.5.1.86"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1023\/A:1017532210579"},{"key":"R13","unstructured":"D. Fudenberg and D. K. Levine,\n                      The Theory of Learning in Games\n                      , MIT Press, Cambridge, MA, 1998."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1006\/jeth.1997.2373"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01763253"},{"key":"R16","unstructured":"A. Graham,\n                      Kronecker Products and Matrix Calculus: with Applications\n                      , Ellis Horwood, Chichester, UK, 1981."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1287\/moor.25.4.591.12120"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01737560"},{"key":"R19","unstructured":"M. Huang, P. E. Caines, and R. P. Malham\u00e9,\n                      Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and Nash equilibrium solutions\n                      , in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, 2003, pp. 98\u2013103"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.904450"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"M. Huang, R. P. Malham\u00e9, and P. E. Caines,\n                      Nash equilibria for large-population linear stochastic systems of weakly coupled agents\n                      , in Analysis, Control and Optimization of Complex Dynamic Systems, E. K. Boukas and R. P. Malham\u00e9, eds., Springer, New York, 2005, pp. 215\u2013252.","DOI":"10.1007\/0-387-25477-3_9"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.4310\/CIS.2006.v6.n3.a5"},{"key":"R23","unstructured":"T. Kailath,\n                      Linear Systems\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1980."},{"key":"R24","doi-asserted-by":"crossref","unstructured":"M. A. Khan and Y. Sun,\n                      Non-cooperative games with many players\n                      , in Handbook of Game Theory with Economic Applications, Vol. 3, R. J. Aumann and S. Hart, eds., Elsevier Science, North-Holland, Amsterdam, 2002, pp. 1761\u20131808.","DOI":"10.1016\/S1574-0005(02)03009-6"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0531(84)90145-5"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.09.019"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.09.018"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1007\/s11537-007-0657-8"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/0309018"},{"key":"R30","unstructured":"J. Maynard Smith,\n                      Evolution and the Theory of Games\n                      , Cambridge University Press, Cambridge, UK, 1982."},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1287\/moor.3.4.290"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1287\/moor.13.4.556"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/BF00933600"},{"key":"R34","unstructured":"J. von Neumann and O. Morgenstern,\n                      Theory of Games and Economic Behavior\n                      , 3rd ed., Princeton University Press, Princeton, NJ, 1953."},{"key":"R35","unstructured":"G. Y. Weintraub, C. L. Benkard, and B. Van Roy,\n                      Oblivious equilibrium: A mean field approximation for large-scale dynamic games\n                      , in Advances in Neural Information Processing Systems, MIT Press, Cambridge, MA, 2005. Available online at http:\/\/books.nips.cc."},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.3982\/ECTA6158"},{"key":"R37","doi-asserted-by":"crossref","unstructured":"W. M. Wonham,\n                      Linear Multivariable Control: A Geometric Approach\n                      , 2nd ed., Springer-Verlag, New York, 1979.","DOI":"10.1007\/978-1-4684-0068-7"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012901391925"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080735370","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:40:56Z","timestamp":1787316056000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080735370"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":38,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080735370"],"URL":"https:\/\/doi.org\/10.1137\/080735370","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}