Interpopulation and Intrapopulation Linear Quadratic Differential Games

We studymultipopulation game problems consisting of both interpopulation and intrapopulation strategic interactions. In the proposed model, the intrapopulation dynamics is given by a stochastic differential equation (SDE) and the strategic interaction occurs among agents who are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">homogeneous</i> within the same population. A stochastic aggregative game takes place in the intrapopulation game problem. The interpopulation dynamics is given by the aggregative behavior of each population and an ordinary differential equation (ODE). The other strategic interaction occurs among different <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">heterogeneous</i> populations in either a noncooperative 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 provideconditions to compute the solution by means of dynamic programming, and providesemi-explicit solutions for the linear-quadratic case by solving the emerging Hamilton–Jacobi–Bellman partial differential equation (PDE) and postulating an appropriate ansatz for the value functional. A numerical example illustrates the presented results.