  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. 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 $+\infty$, thus illustrating in our specific setting the general
results of [8].
