A tale of a Principal and many many Agents
Romuald Elie, Thibaut Mastrolia, Dylan Possama\"i

TL;DR
This paper analyzes a complex moral hazard problem involving infinitely many agents and a principal, using advanced stochastic differential equations and control theory to find explicit solutions and demonstrate convergence to mean-field limits.
Contribution
It introduces a novel approach to solving a mean-field principal-agent problem using FBSDEs and explicit solutions beyond linear-quadratic cases.
Findings
Explicit solutions in special cases beyond LQ framework
Convergence of N-agent optimal contracts to mean-field limit
Application of FBSDE and control techniques to principal-agent models
Abstract
In this paper, we investigate a moral hazard problem in finite time with lumpsum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the meanfield 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 McKeanVlasov SDEs. We review one general approache to tackle it, introduced recently in [1, 43, 44, 45, 46] using dynamic programming and HamiltonJacobiBellman (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 linearquadratic framework. We finally show in our examples that the optimal contract in…
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