Constrained deterministic leader-follower mean field control

We consider a mean field game among a large population of noncooperative agents divided into two categories: leaders and followers. Each agent is subject to heterogeneous convex constraints and minimizes a quadratic cost function; the cost of each leader is affected by the leaders' aggregate strategy, while the cost of each follower is affected by both the leaders' and followers' aggregate strategy. We propose a decentralized scheme in which the agents update their strategies optimally with respect to a global incentive signal, possibly different for leaders and followers, broadcast by a central coordinator. We propose several incentive update rules that, under different conditions on the problem data, are guaranteed to steer the population to an ε-Nash equilibrium, with ε decreasing linearly to zero as the number of players increases. We illustrate our theoretical results on a demand-response program between electricity consumers and producers in the day-ahead market.