Inter- and Intra-Population Linear Quadratic Differential Games

We study multi-population game problems consisting of both inter-population and intra-population strategic interactions. In the proposed model, the intra-population dynamics is given by a Stochastic Differential Equation (SDE) and the strategic interaction occurs among agents who are homogeneous within the same population. A stochastic aggregative game takes place in the intra-population game problem. The inter-population dynamics is given by the aggregative behavior of each population and an Ordinary Differential Equation (ODE). The other strategic interaction occurs among different heterogeneous populations in an either non-cooperative or cooperative way. In addition, the interaction for the different populations is performed in a distributed manner over a graph that might also be considered time-varying. We provide conditions to compute the solution by means of dynamic programming, and provide semi-explicit solutions for the linear-quadratic case by solving the emerging Hamilton-Jacobi-Bellman PDE and postulating an appropriate ansatz for the value functional. A numerical example illustrates the presented results.