  We use the recently developed probabilistic analysis of mean field games with
finitely many states in the weak formulation, to set-up a principal / agent
contract theory model where the principal faces a large population of agents
interacting in a mean field manner. We reduce the problem to the optimal
control of dynamics of the McKean-Vlasov type, and we solve this problem
explicitly in a special case reminiscent of the linear - quadratic mean field
game models. The paper concludes with a numerical example demonstrating the
power of the results when applied to a simple example of epidemic containment.
