{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:43:00Z","timestamp":1772296980710,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T00:00:00Z","timestamp":1533081600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Math Optim"],"published-print":{"date-parts":[[2021,2]]},"DOI":"10.1007\/s00245-018-9510-0","type":"journal-article","created":{"date-parts":[[2018,8,1]],"date-time":"2018-08-01T10:15:53Z","timestamp":1533118553000},"page":"51-82","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition"],"prefix":"10.1007","volume":"83","author":[{"given":"Diogo A.","family":"Gomes","sequence":"first","affiliation":[]},{"given":"Jo\u00e3o","family":"Sa\u00fade","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,8,1]]},"reference":[{"key":"9510_CR1","doi-asserted-by":"publisher","unstructured":"Achdou, Y.: Finite difference methods for mean field games. In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Mathematics, vol. 2074, pp. 1\u201347. Springer, Heidelberg (2013). https:\/\/doi.org\/10.1007\/978-3-642-36433-4_1","DOI":"10.1007\/978-3-642-36433-4_1"},{"issue":"1","key":"9510_CR2","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1137\/100790069","volume":"50","author":"Y Achdou","year":"2012","unstructured":"Achdou, Y., Camilli, F., Capuzzo-Dolcetta, I.: Mean field games: numerical methods for the planning problem. SIAM J. Control Optim. 50(1), 77\u2013109 (2012)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"9510_CR3","doi-asserted-by":"publisher","first-page":"1136","DOI":"10.1137\/090758477","volume":"48","author":"Y Achdou","year":"2010","unstructured":"Achdou, Y., Capuzzo-Dolcetta, I.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136\u20131162 (2010)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"9510_CR4","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1142\/S0218202517400036","volume":"27","author":"Y Achdou","year":"2017","unstructured":"Achdou, Y., Cirant, M., Bardi, M.: Mean field games models of segregation. Math. Models Methods Appl. Sci. 27(1), 75\u2013113 (2017)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"2","key":"9510_CR5","doi-asserted-by":"publisher","first-page":"197","DOI":"10.3934\/nhm.2012.7.197","volume":"7","author":"Y Achdou","year":"2012","unstructured":"Achdou, Y., Perez, V.: Iterative strategies for solving linearized discrete mean field games systems. Netw. Heterog. Media 7(2), 197\u2013217 (2012)","journal-title":"Netw. Heterog. Media"},{"issue":"4","key":"9510_CR6","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1007\/s13235-016-0203-5","volume":"7","author":"N Al-Mulla","year":"2016","unstructured":"Al-Mulla, N., Ferreira, R., Gomes, D.: Two numerical approaches to stationary mean-field games. Dyn. Games Appl. 7(4), 657\u2013682 (2016)","journal-title":"Dyn. Games Appl."},{"issue":"4","key":"9510_CR7","first-page":"449","volume":"8","author":"R Basna","year":"2014","unstructured":"Basna, R., Hilbert, A., Kolokoltsov, V.N.: An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces. Commun. Stoch. Anal. 8(4), 449\u2013468 (2014)","journal-title":"Commun. Stoch. Anal."},{"issue":"3","key":"9510_CR8","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1111\/boer.12024","volume":"67","author":"D Besancenot","year":"2015","unstructured":"Besancenot, D., Dogguy, H.: Paradigm shift: a mean-field game approach. Bull. Econ. Res. 67(3), 289\u2013302 (2015). https:\/\/doi.org\/10.1111\/boer.12024","journal-title":"Bull. Econ. Res."},{"issue":"2","key":"9510_CR9","doi-asserted-by":"publisher","first-page":"801","DOI":"10.1137\/16M1095615","volume":"56","author":"LM Brice\u00f1o Arias","year":"2018","unstructured":"Brice\u00f1o Arias, L.M., Kalise, D., Silva, F.J.: Proximal methods for stationary mean field games with local couplings. SIAM J. Control Optim. 56(2), 801\u2013836 (2018). https:\/\/doi.org\/10.1137\/16M1095615","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"9510_CR10","doi-asserted-by":"publisher","first-page":"279","DOI":"10.3934\/nhm.2012.7.279","volume":"7","author":"P Cardaliaguet","year":"2012","unstructured":"Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games. Netw. Heterog. Media 7(2), 279\u2013301 (2012)","journal-title":"Netw. Heterog. Media"},{"issue":"5","key":"9510_CR11","doi-asserted-by":"publisher","first-page":"3558","DOI":"10.1137\/120904184","volume":"51","author":"P Cardaliaguet","year":"2013","unstructured":"Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51(5), 3558\u20133591 (2013)","journal-title":"SIAM J. Control Optim."},{"issue":"1","key":"9510_CR12","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1137\/120902987","volume":"52","author":"E Carlini","year":"2014","unstructured":"Carlini, E., Silva, F.J.: A fully discrete semi-Lagrangian scheme for a first order mean field game problem. SIAM J. Numer. Anal. 52(1), 45\u201367 (2014)","journal-title":"SIAM J. Numer. Anal."},{"issue":"9","key":"9510_CR13","doi-asserted-by":"publisher","first-page":"4269","DOI":"10.3934\/dcds.2015.35.4269","volume":"35","author":"E Carlini","year":"2015","unstructured":"Carlini, E., Silva, F.J.: A semi-Lagrangian scheme for a degenerate second order mean field game system. Discret. Contin. Dyn. Syst. 35(9), 4269\u20134292 (2015)","journal-title":"Discret. Contin. Dyn. Syst."},{"issue":"1","key":"9510_CR14","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/j.jmaa.2014.02.044","volume":"418","author":"R Ferreira","year":"2014","unstructured":"Ferreira, R., Gomes, D.: On the convergence of finite state mean-field games through $$\\Gamma $$-convergence. J. Math. Anal. Appl. 418(1), 211\u2013230 (2014)","journal-title":"J. Math. Anal. Appl."},{"key":"9510_CR15","unstructured":"Ferreira, R., Gomes, D.: Existence of weak solutions for stationary mean-field games through variational inequalities. Preprint (2016)"},{"issue":"2","key":"9510_CR16","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1016\/j.matpur.2009.10.010","volume":"93","author":"D Gomes","year":"2010","unstructured":"Gomes, D., Mohr, J., Souza, R.R.: Discrete time, finite state space mean field games. J. Math. Pures Appl. 93(2), 308\u2013328 (2010)","journal-title":"J. Math. Pures Appl."},{"issue":"1","key":"9510_CR17","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s00245-013-9202-8","volume":"68","author":"D Gomes","year":"2013","unstructured":"Gomes, D., Mohr, J., Souza, R.R.: Continuous time finite state mean-field games. Appl. Math. Optim. 68(1), 99\u2013143 (2013)","journal-title":"Appl. Math. Optim."},{"key":"9510_CR18","doi-asserted-by":"crossref","unstructured":"Gomes, D., Pimentel, E., Voskanyan, V.: Regularity theory for mean-field game systems. Springer Briefs in Mathematics. Springer, Cham (2016)","DOI":"10.1007\/978-3-319-38934-9"},{"issue":"2","key":"9510_CR19","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1007\/s13235-013-0099-2","volume":"4","author":"D Gomes","year":"2014","unstructured":"Gomes, D., Sa\u00fade, J.: Mean field games models\u2014a brief survey. Dyn. Games Appl. 4(2), 110\u2013154 (2014)","journal-title":"Dyn. Games Appl."},{"key":"9510_CR20","doi-asserted-by":"crossref","unstructured":"Gomes, D., Velho, R.M., Wolfram, M.-T.: Dual two-state mean-field games. In: Proceedings CDC 2014 (2014)","DOI":"10.1109\/CDC.2014.7039803"},{"key":"9510_CR21","doi-asserted-by":"crossref","unstructured":"Gomes, D., Velho, R.M., Wolfram, M.-T.: Socio-economic applications of finite state mean field games. In: Proceedings of the Royal Society A, Bd. 372(2028(S.)) (2014)","DOI":"10.1098\/rsta.2013.0405"},{"issue":"2","key":"9510_CR22","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/s00245-014-9280-2","volume":"72","author":"O Gu\u00e9ant","year":"2015","unstructured":"Gu\u00e9ant, O.: Existence and uniqueness result for mean field games with congestion effect on graphs. Appl. Math. Optim. 72(2), 291\u2013303 (2015). https:\/\/doi.org\/10.1007\/s00245-014-9280-2","journal-title":"Appl. Math. Optim."},{"key":"9510_CR23","unstructured":"Gu\u00e9ant, O.: From infinity to one: the reduction of some mean field games to a global control problem. Preprint (2011)"},{"issue":"9","key":"9510_CR24","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1109\/TAC.2007.904450","volume":"52","author":"M Huang","year":"2007","unstructured":"Huang, M., Caines, P.E., Malham\u00e9, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $$\\epsilon $$-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560\u20131571 (2007)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"9510_CR25","doi-asserted-by":"crossref","first-page":"221","DOI":"10.4310\/CIS.2006.v6.n3.a5","volume":"6","author":"M Huang","year":"2006","unstructured":"Huang, M., Malham\u00e9, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221\u2013251 (2006)","journal-title":"Commun. Inf. Syst."},{"issue":"1","key":"9510_CR26","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/s13235-015-0175-x","volume":"7","author":"VN Kolokoltsov","year":"2017","unstructured":"Kolokoltsov, V.N., Malafeyev, O.A.: Mean-field-game model of corruption. Dyn. Games Appl. 7(1), 34\u201347 (2017)","journal-title":"Dyn. Games Appl."},{"issue":"9","key":"9510_CR27","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1016\/j.crma.2006.09.019","volume":"343","author":"J-M Lasry","year":"2006","unstructured":"Lasry, J.-M., Lions, P.-L.: Jeux \u00e0 champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619\u2013625 (2006)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"10","key":"9510_CR28","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1016\/j.crma.2006.09.018","volume":"343","author":"J-M Lasry","year":"2006","unstructured":"Lasry, J.-M., Lions, P.-L.: Jeux \u00e0 champ moyen. II. Horizon fini et contr\u00f4le optimal. C. R. Math. Acad. Sci. Paris 343(10), 679\u2013684 (2006)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"1","key":"9510_CR29","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1137\/17M1125960","volume":"50","author":"A M\u00e9sz\u00e1ros","year":"2018","unstructured":"M\u00e9sz\u00e1ros, A., Silva, F.J.: On the variational formulation of some stationary second-order mean field games systems. SIAM J. Math. Anal. 50(1), 1255\u20131277 (2018). https:\/\/doi.org\/10.1137\/17M1125960","journal-title":"SIAM J. Math. Anal."},{"issue":"6","key":"9510_CR30","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1016\/j.matpur.2015.07.008","volume":"104","author":"AR M\u00e9sz\u00e1ros","year":"2015","unstructured":"M\u00e9sz\u00e1ros, A.R., Silva, F.J.: A variational approach to second order mean field games with density constraints: the stationary case. J. Math. Pures Appl. (9) 104(6), 1135\u20131159 (2015)","journal-title":"J. Math. Pures Appl. (9)"}],"container-title":["Applied Mathematics &amp; Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00245-018-9510-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00245-018-9510-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00245-018-9510-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,28]],"date-time":"2022-08-28T15:21:00Z","timestamp":1661700060000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00245-018-9510-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,8,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["9510"],"URL":"https:\/\/doi.org\/10.1007\/s00245-018-9510-0","relation":{},"ISSN":["0095-4616","1432-0606"],"issn-type":[{"value":"0095-4616","type":"print"},{"value":"1432-0606","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,8,1]]},"assertion":[{"value":"1 August 2018","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}