Sequential decomposition of dynamic games with asymmetric information and dependent states

We consider a finite-horizon dynamic game with asymmetric information with $N$ selfish players, where there exists an underlying state of the system that is a controlled Markov process, controlled by players' actions. In each period, a player makes a common observation of the state together with all the other players, and a private observation, and gets an instantaneous reward which is a function of the state and everyone's' actions. The players' private observations are conditionally independent across time, conditioned on the system state and players' previous actions, however, they are potentially correlated among players in each period. This observation model includes the case when players observe their rewards at the end of each period. We first solve the team version of this problem where users are cooperative i.e. have the same objective. Using these results as motivation for the definition of information state, we present a sequential decomposition methodology to compute \emph{structured perfect Bayesian equilibria} (SPBE) of this game, introduced in~[1]. In general, these equilibria exhibit \textit{signaling} behavior, i.e. players' actions reveal part of their private information that is payoff relevant to other users.