A tale of a Principal and many many Agents

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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 one general approache to tackle it, introduced recently in [1, 43, 44, 45, 46] using dynamic programming and Hamilton$-$Jacobi$-$Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle, which follows [10].
• We solve completely and explicitly the problem in special cases, going beyond the usual linear$-$quadratic framework.