{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:35:05Z","timestamp":1787326505246,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In this paper we formulate and analyze an $N$-player stochastic game of the classical fuel follower problem and its mean field game (MFG) counterpart. For the $N$-player game, we obtain the Nash equilibrium (NE) explicitly by deriving and analyzing a system of Hamilton--Jacobi--Bellman equations and by establishing the existence of a unique strong solution to the associated Skorokhod problem on an unbounded polyhedron with an oblique reflection. For the MFG, we derive a bang-bang type NE under some mild technical conditions and by the viscosity solution approach. We also show that this solution is an $\\epsilon$-NE to the $N$-player game, with $\\epsilon =O(\\frac{1}{\\sqrt{N}})$. The $N$-player game and the MFG differ in that the NE for the former is state dependent while the NE for the latter is a threshold-type bang-bang policy where the threshold is state independent. Our analysis shows that the NE for a stationary MFG may not be the NE for the corresponding MFG.<\/jats:p>","DOI":"10.1137\/17m1159531","type":"journal-article","created":{"date-parts":[[2019,2,19]],"date-time":"2019-02-19T11:11:53Z","timestamp":1550574713000},"page":"659-692","source":"Crossref","is-referenced-by-count":31,"title":["Stochastic Games for Fuel Follower Problem: $N$ versus Mean Field Game"],"prefix":"10.1137","volume":"57","author":[{"given":"Xin","family":"Guo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Renyuan","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,2,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1214\/009117906000000359"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.3934\/nhm.2012.7.243"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/140951795"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1080\/17442508008833156"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004637022496"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0819-4"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1214\/105051606000000556"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1079021464"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2008.03.001"},{"key":"atypb10","unstructured":"P. Cardaliaguet, F. Delarue, J.M. Lasry, and P.L. Lions,\n                      The Master Equation and the Convergence Problem in Mean Field Games\n                      , preprint,arXiv:1509.02505, 2015."},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"R. Carmona,\n                      Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications\n                      , Financ. Math. 1, SIAM, Philadelphia, 2016.","DOI":"10.1137\/1.9781611974249"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2015.v13.n4.a4"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"S. Cohen and R. Elliott,\n                      Stochastic Calculus and Applications\n                      , Birkh\u00e4user, Basel, 2015.","DOI":"10.1007\/978-1-4939-2867-5"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/1140001"},{"key":"atypb15","unstructured":"T. De Angelis and G. Ferrari,\n                      Stochastic Non-Zero-Sum Games: A New Connection Between Singular Control and Optimal Stopping\n                      , preprint,arXiv:1601.05709, 2016."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176989415"},{"key":"atypb17","first-page":"205","volume":"36","author":"Engelbert H.","year":"1991","journal-title":"Stochastics"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1080\/03605307908820103"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/17M1123742"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"X. Guo and J. S. Lee,\n                      Stochastic Games and Mean Field Games with Singular Controls\n                      , preprint, 2018.","DOI":"10.2139\/ssrn.2932277"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2004.12.002"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2014.06.011"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176992259"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1214\/13-AAP986"},{"key":"atypb25","unstructured":"Y. Hu, B. \u00d8ksendal, and A. Sulem,\n                      Singular Mean-Field Control Games with Applications to Optimal Harvesting and Investment Problems\n                      , preprint,arXiv:1406.1863, 2014."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.904450"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.4310\/CIS.2006.v6.n3.a5"},{"key":"atypb28","unstructured":"R. Hynd,\n                      Partial Differential Equations with Gradient Constraints Arising in the Optimal Control of Singular Stochastic Processes\n                      , Ph.D. dissertation, University of California, Berkeley, 2010."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.2307\/1426435"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1142\/9789814383318_0006"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/0323028"},{"key":"atypb32","unstructured":"I. Karatzas and S. Shreve,\n                      Brownian Motion and Stochastic Calculus\n                      , Grad. Texts in Math. 113, Springer, New York, 2012."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998347535"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2014.0700"},{"key":"atypb35","unstructured":"D. Lacker and T. Zariphopoulou,\n                      Mean Field and N-Agent Games for Optimal Investment Under Relative Performance Criteria\n                      , preprint,arXiv:1703.07685, 2017."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/s11537-007-0657-8"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012903423715"},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"M. Nutz and Y. Zhang,\n                      A Mean Field Competition\n                      , preprint,arXiv:1708.01308, 2017.","DOI":"10.2139\/ssrn.3013429"},{"key":"atypb39","first-page":"265","volume":"16","author":"Shreve S.","year":"1991","journal-title":"Appl. Stoch. Anal."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/1101022"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/0327047"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160380405"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/BF00320328"},{"key":"atypb44","unstructured":"L. Zhang,\n                      The Relaxed Stochastic Maximum Principle in the Mean-Field Singular Controls\n                      , preprint,arXiv:1202.4129, 2012."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1159531","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:32:18Z","timestamp":1787322738000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1159531"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1159531"],"URL":"https:\/\/doi.org\/10.1137\/17m1159531","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}