Master equation of discrete time graphon mean field games and teams

In this paper, we present a sequential decomposition algorithm equivalent of Master equation to compute GMFE of GMFG and graphon optimal Markovian policies (GOMPs) of graphon mean field teams (GMFTs). We consider a large population of players sequentially making strategic decisions where the actions of each player affect their neighbors which is captured in a graph, generated by a known graphon. Each player observes a private state and also a common information as a graphon mean-field population state which represents the empirical networked distribution of other players' types. We consider non-stationary population state dynamics and present a novel backward recursive algorithm to compute both GMFE and GOMP that depend on both, a player's private type, and the current (dynamic) population state determined through the graphon. Each step in computing GMFE consists of solving a fixed-point equation, while computing GOMP involves solving for an optimization problem. We provide conditions on model parameters for which there exists such a GMFE. Using this algorithm, we obtain the GMFE and GOMP for a specific security setup in cyber physical systems for different graphons that capture the interactions between the nodes in the system.