On the Master Equation for Linear Quadratic Graphon Mean Field Games

In this work a linear quadratic instance of Graphon Mean-Field Games (GMFGs) is analysed. Such games involve an asymptotically infinite population of agents, distributed over a very large scale network which itself is asymptotically infinite. The linear dynamics of each agent together with its quadratic running and terminal cost functions depend upon non-uniform averages (i.e. local and global mean fields) of the states of all other agents in the network system. In the infinite limit of the population and the network, the agent's dynamics and costs are functions of the family of local mean fields distributed at the nodes of the infinite network. Moreover, the limiting infinite networks are modelled by graphons which are symmetric measurable functions defined on the unit square. Specifically, Linear Quadratic Graphon Mean Field Games model the idea of clustering for populations of agents at the nodes of the very large scale network. First, using a probabilistic approach, we characterize the solutions of the Linear Quadratic Graphon Mean Field Games with solutions to coupled Forward Backward Stochastic Differential Equations (FBSDEs) of McKean-Vlasov type. We next deduce the existence of the so-called Master field, which allows for the decoupling of these FBSDEs. Finally, we derive the infinite dimensional Partial Differential Equation (PDE), so-called Master Equation, for which the Master Field is a solution.