Linear Quadratic Graphon Field Games

Linear quadratic graphon field games (LQ-GFGs) are defined to be LQ games which involve a large number of agents that are weakly coupled via a weighted undirected graph on which each node represents an agent. The links of the graph correspond to couplings between the agents' dynamics, as well as between the individual cost functions, which each agent attempts to minimize. We formulate limit LQ-GFG problems based on the assumption that these graphs lie in a sequence which converges to a limit graphon. First, under a finite-rank assumption on the limit graphon, the existence and uniqueness of solutions to the formulated limit LQ-GFG problem is established. Second, based upon the solutions to the limit LQ-GFG problem, epsilon-Nash equilibria are constructed for the corresponding game problems with a very large but finite number of players. This result is then generalized to the case with random initial conditions. It is to be noted that LQ-GFG problems are distinct from the class of graphon mean field game (GMFG) problems where a population is hypothesized to be associated with each node of the graph [Caines and Huang CDC 2018, 2019].