Stackelberg Mean Field Games: convergence and existence results to the problem of principal with multiple agents in competition

In a situation of moral hazard, this paper investigates the problem of Principal with n Agents when the number of Agents n goes to infinity. There is competition between the Agents expressed by the fact that they optimize their utility functions through a Nash equilibrium criterion. Each Agent is offered by the Principal a contract which is divided into a Markovian part involving the state/production of the Agent and a non--Markovian part involving the states/productions of all the other Agents. The Agents are in interactions. By the help of the mean field games theory, we are able to formulate an appropriate limit problem involving a Principal with a representative Agent. We start by solving the problem of both the Principal and the representative Agent in this limit problem. Then, when n goes to infinity, we show that the problem of Principal with n Agents converges to the limit problem of Principal with a representative Agent. A notable result is that, despite allowing a general type of contracts, it is approximately optimal for the Principal to offer contracts to the n Agents that are maps of the empirical distribution of states/productions and controls of the Agents