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Therefore, the transition probability and the one-stage cost function of each agent depend linearly on the mean-field term, which is the key distinction between classical mean-field games and linear mean-field games. Under mild assumptions, we show that the policy obtained from infinite population equilibrium is [Formula: see text]-Nash when the number of agents N is sufficiently large, where [Formula: see text] is an explicit function of N. Then, using the linear programming formulation of Markov decision processes (MDPs) and the linearity of the transition probability in the mean-field term, we formulate the game in the infinite population limit as a generalized Nash equilibrium problem (GNEP) and establish an algorithm for computing equilibrium with a convergence guarantee.<\/jats:p>","DOI":"10.1287\/moor.2023.0148","type":"journal-article","created":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T08:54:41Z","timestamp":1739782481000},"page":"358-397","source":"Crossref","is-referenced-by-count":1,"title":["Linear Mean-Field Games with Discounted Cost"],"prefix":"10.1287","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2677-7366","authenticated-orcid":false,"given":"Naci","family":"Saldi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bilkent University, Cankaya, 06800 Ankara, Turkey"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jet.2013.07.002"},{"key":"B2","volume-title":"Infinite Dimensional Analysis","author":"Aliprantis CD","year":"2006","edition":"3"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2020.104744"},{"key":"B4","first-page":"89","volume":"13","author":"Anahtarci B","year":"2022","journal-title":"Dynam. 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