Master equation of discrete-time Stackelberg mean field games with multiple leaders

In this paper, we consider a discrete-time Stackelberg graphon mean field game with a finite number of leaders, a finite number of major followers and an infinite number of minor followers. The leaders and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leaders and the followers sequentially make strategic decisions where each follower's actions affect her neighbors, which is captured in a graph generated by a known graphon, however, the leaders' actions affect everyone. The leaders are of "Stackelberg" kind which means each of them commits to a dynamic policy and all the followers(both major and minor) best respond to that policy and each other. Knowing that the minor followers would best respond (in the sense of a mean-field game) while the major followers will best respond (in the sense of NAsh) based on their policies, each leader chooses a policy that maximizes her reward knowing that other leader's are doing the same. We refer to the resulting outcome as a Stackelberg Graphon Mean Field Equilibrium with multiple leaders (SGMFE-ML). In this paper, we provide a master equation of this game that allows one to compute all SGMFE-ML. We further extend this notion to the case when there are an infinite number of leaders.