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A tale of a principal and many many agents

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 for short), we are able to rewrite the Principal's problem as a control problem of McKean–Vlasov SDEs. We review two general approaches to tackle it: the first one introduced recently in [2, 66, 67, 68, 69] using dynamic programming and Hamilton–Jacobi– Bellman (HJB for short) equations, the second based on the stochastic Pontryagin maximum principle, which follows [16]. 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' model converges to the mean–field optimal contract when the number of agents goes to +∞, this illustrating in our specific setting the general results of [12].
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Submitted on : Thursday, March 2, 2017 - 3:20:21 PM
Last modification on : Tuesday, January 18, 2022 - 3:23:27 PM
Long-term archiving on: : Wednesday, May 31, 2017 - 2:39:00 PM

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Romuald Elie, Thibaut Mastrolia, Dylan Possamaï. A tale of a principal and many many agents. Mathematics of Operations Research, INFORMS, 2019, 44 (2), pp.440-467. ⟨10.1287/moor.2018.0931⟩. ⟨hal-01481390⟩

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