{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T07:51:41Z","timestamp":1725695501086},"reference-count":39,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2019,5]]},"abstract":" In this paper, we investigate a moral hazard problem in finite time with lump-sum and continuous payments, involving infinitely many agents with mean-field type interactions, hired by one principal. By reinterpreting the mean-field game faced by each agent in terms of a mean-field forward-backward stochastic differential equation (FBSDE), we are able to rewrite the principal\u2019s problem as a control problem of the McKean-Vlasov stochastic differential equations. We review one general approach to tackling it, introduced recently using dynamic programming and Hamilton-Jacobi-Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle. We solve completely and explicitly the problem in special cases, going beyond the usual linear-quadratic framework. We finally show in our examples that the optimal contract in the N-players\u2019 model converges to the mean-field optimal contract when the number of agents goes to +\u221e. <\/jats:p>","DOI":"10.1287\/moor.2018.0931","type":"journal-article","created":{"date-parts":[[2018,9,24]],"date-time":"2018-09-24T18:47:28Z","timestamp":1537814848000},"page":"440-467","source":"Crossref","is-referenced-by-count":45,"title":["A Tale of a Principal and Many, Many Agents"],"prefix":"10.1287","volume":"44","author":[{"given":"Romuald","family":"Elie","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris\u2013Est Marne\u2013la\u2013Vall\u00e9e, Champs\u2013sur\u2013Marne 77454 Marne\u2013la\u2013Vall\u00e9e CEDEX 2, France"}]},{"given":"Thibaut","family":"Mastrolia","sequence":"additional","affiliation":[{"name":"CMAP, \u00c9cole Polytechnique, Universit\u00e9 Paris Saclay, 91128 Palaiseau, France"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-9364-0124","authenticated-orcid":false,"given":"Dylan","family":"Possama\u00ef","sequence":"additional","affiliation":[{"name":"Columbia University, New York, New York 10027"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/7118"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-8508-7"},{"key":"B3","volume-title":"Contract Theory","author":"Bolton P","year":"2005"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2009.05.002"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1214\/08-AOP442"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1214\/15-AOP1076"},{"issue":"68","key":"B9","first-page":"1","volume":"18","author":"Carmona R","year":"2013","journal-title":"Electronic Comm. Probab."},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1214\/14-AOP946"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1214\/14-AAP1020"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2015.v13.n4.a4"},{"issue":"4","key":"B14","first-page":"507","volume":"20","author":"Chiang TS","year":"1994","journal-title":"Soochow J. Math."},{"key":"B15","volume-title":"Contract Theory in Continuous-Time Models","author":"Cvitani\u0107 J","year":"2012"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-017-0344-4"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2016.2493"},{"key":"B18","first-page":"328","author":"Dellacherie C","year":"1981","journal-title":"S\u00e9minaire de probabilit\u00e9s de Strasbourg"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0531(84)90045-0"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00022"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1086\/261153"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.2307\/3003457"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.2307\/1913238"},{"key":"B28","doi-asserted-by":"publisher","DOI":"10.1007\/s11424-007-9015-4"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.4310\/CIS.2006.v6.n3.a5"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2008.00352.x"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1515\/9781400829453"},{"key":"B32","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.09.019"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2006.09.018"},{"key":"B34","doi-asserted-by":"publisher","DOI":"10.1007\/s11537-007-0657-8"},{"key":"B35","first-page":"1","volume":"1","author":"L\u00e9pingle D","year":"1978","journal-title":"Publications des s\u00e9minaires de math\u00e9matiques et informatique de Rennes"},{"key":"B36","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392299"},{"key":"B37","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(89)90062-1"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.2307\/2297432"},{"issue":"7","key":"B43","volume":"1","author":"Pham H","year":"2016","journal-title":"Probab., Uncertainty Quant. Risk"},{"key":"B45","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-016-9386-9"},{"key":"B46","doi-asserted-by":"publisher","DOI":"10.1137\/16M1071390"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-06400-9"},{"key":"B48","doi-asserted-by":"publisher","DOI":"10.1006\/jeth.1993.1072"},{"key":"B49","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-1889(96)00944-X"},{"key":"B50","doi-asserted-by":"publisher","DOI":"10.2307\/2556015"},{"key":"B52","doi-asserted-by":"publisher","DOI":"10.1214\/17-AOP1190"}],"container-title":["Mathematics of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubsonline.informs.org\/doi\/pdf\/10.1287\/moor.2018.0931","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T20:42:26Z","timestamp":1680468146000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubsonline.informs.org\/doi\/10.1287\/moor.2018.0931"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,5]]}},"alternative-id":["10.1287\/moor.2018.0931"],"URL":"https:\/\/doi.org\/10.1287\/moor.2018.0931","relation":{},"ISSN":["0364-765X","1526-5471"],"issn-type":[{"value":"0364-765X","type":"print"},{"value":"1526-5471","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5]]}}}