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We conclude the main theoretical portion of the paper by establishing a large deviation principle for empirical measures associated with the asymptotic Nash equilibria. Then, we contrast the asymptotic Nash equilibria using an example. We solve the MFG system directly and numerically solve the ergodic master equation by adapting the deep Galerkin method. We use these results to derive the strategies of the asymptotic Nash equilibria and compare them. Finally, we derive an explicit form for the rate functions in dimension two.<\/jats:p>","DOI":"10.1287\/moor.2024.0478","type":"journal-article","created":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T10:41:28Z","timestamp":1746787288000},"page":"1139-1173","source":"Crossref","is-referenced-by-count":4,"title":["Asymptotic Nash Equilibria of Finite-State Ergodic Markovian Mean Field Games"],"prefix":"10.1287","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9211-7956","authenticated-orcid":false,"given":"Asaf","family":"Cohen","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ethan","family":"Zell","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2016.06.004"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1137\/140951795"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1137\/17M113887X"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2014.11.005"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1287\/12-SSY064"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2019.12.1397"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1007\/s00030-020-00628-w"},{"key":"B8","volume-title":"The Master Equation and the Convergence Problem in Mean Field Games,","volume":"201","author":"Cardaliaguet P","year":"2019"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1137\/19M1274377"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-018-9488-7"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2018.12.002"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1017\/S0334270000012492"},{"key":"B13","unstructured":"Cohen A, Huffman E (2025) Uniform-in-time convergence rates to a nonlinear Markov chain for mean-field interacting jump processes. 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