Principal-agent problem with multiple principals

We consider a moral hazard problem with multiple principals in a continuous-time model. The agent can only work exclusively for one principal at a given time, so faces an optimal switching problem. Using a randomized formulation and techniques from the theory of backward SDEs, we manage to represent the agent's value function and his optimal effort by an Ito process. This representation further helps to solve the principals' problem in case we have infinite number of principals in the sense of mean field game. Finally, to justify the mean field formulation, we develop the so-called backward propagation of chaos, which may carry independent interest itself.