A Tale of a Principal and Many, Many Agents

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’s 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’ model converges to the mean-field optimal contract when the number of agents goes to +∞.