On constrained mean field control for large populations of heterogeneous agents: Decentralized convergence to Nash equilibria

We consider mean field games in a large population of heterogeneous agents subject to convex constraints and coupled by a quadratic cost, which depends on the average population behavior. The problem of steering such population to a Nash equilibrium is usually addressed in the (mean field control) literature by formulating an iterative game between the agents and a central coordinator, that broadcasts at every step the average population behavior. Here we generalize this approach by allowing the central operator to filter such signal using a feedback mapping. We propose different classes of feedback mappings and we derive sufficient conditions guaranteeing convergence to a ε-Nash equilibrium, even for cases when the standard approach fails. We show that the deviation ε of each agent from its optimal cost, decreases at least linearly to zero with the increase of the population size. Contrary to the state of the art, the proposed approach guarantees convergence in the presence of heterogeneous convex constraints for the agents. Finally, we show how these results can be applied to regulate in a decentralized fashion the charging process of a large population of plug-in electric vehicles. Our findings give theoretical support and extend previous literature results.