N-Person Discrete-Time Dynamic Games of Asymmetric Information

This paper considers a class of N-person discrete-time dynamic games with an asymmetric information structure. Each player has private information revealed only to himself, which is modeled as a random variable called the type. Each player aims to find an optimal feedback control policy to reach the desired state while minimizing the control cost without exact knowledge of the system dynamics. Players can form a belief on the unknowns based on the observation of the state trajectory and update it via the Bayesian rule to learn the type value of other players. To deal with the uncertainty caused by the private information, each player forms his control policy under the expectation of the type belief, which forms the perfect Bayesian Nash equilibrium (PBNE). The strong coupling of the type estimation and the control policy establishes no separation principle in our non-classical information structure. In particular, we investigate the linear-quadratic setting, and we obtain generalized Riccati equations and an affine state-feedback PBNE policy. Moreover, we show that the PBNE policy is unique if it exists and is strongly time consistent. Finally, we numerically illustrate the proposed framework with a case study.