# A tale of a Principal and many many Agents

arXiv: https://arxiv.org/abs/1608.05226 · HTML: https://mlsanity.com/paper/1608.05226/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Aug 2016.

## Built on by (most cited first, 7 of 7)

- [Deep Learning for Principal-Agent Mean Field Games](https://mlsanity.com/paper/2110.01127/) (2021)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)
- [Learning Discrete-Time Major-Minor Mean Field Games](https://mlsanity.com/paper/2312.10787/) (2023)
- [Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs](https://mlsanity.com/paper/2410.14788/) (2024)
- [MESOB: Balancing Equilibria & Social Optimality](https://mlsanity.com/paper/2307.07911/) (2023)
- [Steering No-Regret Agents in MFGs under Model Uncertainty](https://mlsanity.com/paper/2503.09309/) (2025)
- [Bounded Rationality Equilibrium Learning in Mean Field Games](https://mlsanity.com/paper/2411.07099/) (2024)

## Abstract (arXiv)

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