A tale of a Principal and many many Agents

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 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].